UNIVERSALITY IN MEAN CURVATURE FLOW NECKPINCHES
ZHOU GANG AND DAN KNOPF
Abstract. We study noncompact surfaces evolving by mean curvature flow.
Without any symmetry assumptions, we prove that any solution that is C 3 close at some time to a standard neck will develop a neckpinch singularity in
finite time, will become asymptotically rotationally symmetric in a space-time
neighborhood of its singular set, and will have a unique tangent flow.
Contents
1. Introduction
2. Preliminaries
3. How the solution evolves
4. The first bootstrap machine
5. Improved estimates for the inner region
6. The second bootstrap machine
7. Improved estimates for the decomposition
8. Proof of the Main Theorem
Appendix A. Proof of Lemma 9
Appendix B. Proof of Lemma 11
Appendix C. Proof of Lemma 12
References
1
4
9
9
14
23
26
30
32
38
44
50
1. Introduction
In this paper, we prove, without imposing any symmetry assumptions, that
any complete noncompact two-dimensional solution of mean curvature flow (mcf)
that is close to a standard round neck at some time will (i) encounter a finitetime singularity, (ii) become asymptotically rotationally symmetric in a space-time
neighborhood of the developing singularity, (iii) satisfy an exact asymptotic profile
in that neighborhood, and (iv) will as a consequence have a unique tangent flow. All
of these statements are made precise below. This result extends our previous work
on mcf singularities: in that work [13], we removed the hypothesis of rotational
symmetry but retained certain discrete symmetry hypotheses that served to fix
the limiting cylinder. The results in this paper, combined with significant work
of others, makes it reasonable to expect that singularities of generic mcf solutions
may be constrained to a small selection of “universal” asymptotic profiles. Before
giving the details, of our results, we sketch the broad outlines of the emerging
picture motivating this expectation.
ZG thanks NSF for support in DMS-1308985. DK thanks NSF for support in DMS-1205270.
1
2
ZHOU GANG AND DAN KNOPF
m+1
One says a smooth one-parameter family Mm
of hypersurfaces moves
t ⊂ R
m
by mcf if at every point x ∈ Mt , one has ∂t x = −Hν, where H is the mean
curvature scalar, and ν is the unit normal at x. Denote the backward heat kernel
at (x0 , t0 ) by ρ(x0 ,t0 ) (x, t) = {4π(t0 − t)}−m/2 exp{−|x − x0 |2 /4(t0 − t)}. For t < t0 ,
Huisken’s mcf monotonicity formula [17] states that
2
Z
Z
(x − x0 )⊥
d
ρ(x ,t ) (x, t) dµ.
ρ(x0 ,t0 ) (x, t) dµ = −
−
Hν
0 0
dt Mm
2(t0 − t)
Mm
t
t
A consequence of this is the following characterization of tangent flows (singularity
models) for Type-I mcf singularities:
Theorem A (Huisken). Given any sequence of parabolic dilations at a Type-I
mcf singularity, there exists a subsequence that converges smoothly to a nonempty
immersed self-similarly shrinking solution.
In important subsequent work, Huisken and Sinestrari prove that any mean
convex solution becomes asymptotically convex at its first singular time [18], and
they develop a complete surgery program for solutions originating from two-convex
m+1
initial data Mm
that are immersed images of compact manifolds of dimen0 ⊂R
sions m ≥ 3 [19]. In the latter work, they show in particular that all singularities
of such solutions are either spherical (Sm ) or neckpinch (R × Sm−1 ) singularities.
Subsequently, Brendle and Huisken [5] extend this result to mcf with surgery of
mean-convex initial surfaces M20 ⊂ R3 . Very recently, Haslhofer and Kleiner [16]
obtain similar results for mcf with surgery, without dimension restrictions, using
shorter proofs that rely on blow-up arguments.
m
m+1
Also recently, given t0 > 0 and a hypersurface
, Colding and MiniR M ⊂R
m
cozzi consider a functional F(x0 ,t0 ) (M ) = Mm ρ(x0 ,t0 ) (·, 0) dµ, and an entropy
λ(Mm ) = sup(x0 ,t0 ) F(x0 ,t0 ) (Mm ). The entropy λ is invariant under dilations and
Euclidean motions, and λ(Mm
t ) is nonincreasing under mcf. Among other results,
they prove a stability property for tangent flows [8].
Theorem B (Colding–Minicozzi). Let Mm ⊂ Rm+1 be a smooth1 complete embedded self-shrinker with polynomial volume growth and without boundary.
If Mm is not equal to any Sk × Rm−k , 0 ≤ k ≤ m, then for any fixed r, there
is a graph Nm over Mm of a function with arbitrarily small C r norm such that
λ(Nm ) < λ(Mm ).
A consequence of Theorem B is that Mm cannot occur as a tangent flow of a
mcf solution whose initial data are any perturbations Nm of Mm . It follows that
spheres and cylinders are the only generic self-similarly shrinking tangent flows.
Left open by these important results is the question of whether singularity
models are independent of subsequence. This has been answered affirmatively by
Schulze [23] (using the Simon–Lojasiewicz inequality) if one tangent flow is a closed,
multiplicity-one, smoothly embedded self-similar shrinker, but the general case remains open. Progress toward resolving this question was recently made by Colding,
Ilmanen, and Minicozzi [7], who prove that if one tangent flow at a singularity is a
1Recall that by Ilmanen’s R3 blowup theorem [21], a compact surface evolving by mcf has
a smooth singularity model at its first singular time. In dimensions 3 ≤ m ≤ 6, Theorem B
also holds if Mm is merely smooth away from a singular set of locally finite (m − 2)-dimensional
Hausdorff measure.
UNIVERSALITY IN MCF NECKPINCHES
3
multiplicity-one generalized cylinder, then every subsequential limit is some generalized cylinder. Since this paper was written, its authors learned of important work
of Colding and Minicozzi [9], who prove that tangent cones are unique for generic
singularities of mcf, and for all singularities of mean-convex mcf. Other recent
progress towards classifying singularity models comes from work of Wang [24], who
proves that there is at most one smooth complete properly embedded self-shrinker
asymptotic at spatial infinity to any regular cone.
In this paper, we show that the tangent cylinders of certain mcf neckpinches
are independent of subsequence by proving that these singularities have unique
asymptotic behavior. With few exceptions (i.e. [11] for logarithmic fast diffusion)
asymptotic analysis results of this nature (e.g. [22] and [10] for logarithmic fast
diffusion; [3], [1], and [2] for Ricci flow; and [4] and [14] for mcf) require a restrictive
hypothesis of rotational symmetry. In [13], with I.M. Sigal, we made partial progress
towards showing uniqueness by removing the hypothesis of rotational symmetry
for mcf neckpinches, retaining only weaker discrete symmetries, and then proving
that neckpinches with these discrete symmetries asymptotically become rotationally
symmetric. (See [13] for precise statements of these assumptions and results.) Here,
we extend that work by removing symmetry assumptions altogether. We show
that neckpinches originating from initial data sufficiently close to a formal solution
develop unique asymptotic profiles, modulo dilations and Euclidean motions.
At least if Ilmanen’s multiplicity conjecture [20] is true, then combining our
work here with the results cited above makes it reasonable to conjecture that mcf
solutions originating from generic initial data M20 ⊂ R3 are constrained to only two
universal asymptotic profiles if they become singular.
Passing to higher dimensions, we strongly conjecture that our main results in
m+1
in all dimensions.
this paper generalize to hypersurface neckpinches Mm
t ⊂ R
Indeed, the strategy of the proof, as outlined in Section 2.3, rests on the fact that
although the linearization of mcf at a cylinder R × S1 is formally unstable, the
unstable eigenmodes are simply “coordinate instabilities.” By properly choosing
seven coordinate parameters, a solution is decomposed into a dominant component,
and a rapidly-decaying component on which the linearization is strictly stable. The
only complications in extending these ideas to dimensions m > 2 are that one
must deal with m + 5 coordinate parameters, and with curvature terms that arise
from commuting covariant derivatives on Sm−1 . The latter, however, are harmless,
because the coordinate parametrization controls the size of the cylinder.
This paper is structured as follows. As in [13], we prove our results by means
of two bootstrap machines. In Section 2, we establish notation, state our main
assumptions, and outline the main ideas introduced here to generalize our earlier
work. In Section 3, we derive the evolution equations for the quantities under
analysis. We construct the first bootstrap machine in Sections 4–5 and the second
in Sections 6–7. We complete the proof of the Main Theorem in Section 8. The
proofs of several supporting technical results appear in the appendices.
We now summarize the main results of this paper, using the following terminology. We say that (x, r, θ) is a cylindrical coordinate system if there exist orthonormal
coordinates (x0 , x1 , x2 ) for R3 such that x = x0 is the cylindrical axis and (r, θ)
are polar coordinates for the (x1 , x2 )-plane. In this notation, the standard cylinder
with axis x is the set {r = 1}, and a normal graph is determined by r = u(x, θ).
4
ZHOU GANG AND DAN KNOPF
Main Theorem. Suppose that a solution of mcf satisfies Assumptions [A1]–[A7]
from Section 2.2 for sufficiently small b0 , c0 .
(i) There exists T < ∞ such that the solution becomes singular at time T .
(ii) There exists a sequence (x, r, θ)n of cylindrical coordinate systems such that
for all times tn−1 ≤ t ≤ tn , the surface can be written as the normal graph of
a positive function u(x, θ, t). Moreover, tn % T as n → ∞, and there exists a
limiting (x, r, θ)n → (x, r, θ)∞ cylindrical coordinate system. In these coordinates,
the solution develops a neckpinch, with u(x, ·, t) = 0 if and only if x = 0 and t = T .
(iii) In the cylindrical coordinate system (x, r, θ)n constructed at time tn , the
solution admits an “optimal coordinate” decomposition
s
2 + bopt (tn )y 2
u(x, θ, tn )
=
+ φopt (y, θ, tn ),
λopt (tn )
1 + 12 bopt (tn )
where y = λopt (tn )−1 x. The parameters λopt (tn ) and bopt (tn ) have the asymptotic
behavior that as tn % T ,
p
λopt (tn ) = {1 + o(1)} T − tn ,
bopt (tn ) = {1 + o(1)}(− log(T − tn ))−1 .
(iv) The solution is asymptotically rotationally symmetric near the singularity —
in the precise technical senses that there exist ε1 , ε2 > 0 such that in each cylindrical
coordinate system (x, r, θ)n at all times t ∈ [tn−1 , tn ], there exist C 1 functions λ,
a, b, and β0 , . . . , β4 of time, with λ(tn ) = {1 + o(1)}λopt (tn ), a(tn ) = {1 + o(1)} 21 ,
b(tn ) = {1 + o(1)}bopt (tn ), and |βk (t)| . (− log(T − t))−2 for k = 0, . . . , 4, such
that the θ-dependent component φ of the solution defined by
r
u
2 + by 2
φ := −
− β0 y − β1 cos θ − β2 sin θ − β3 y cos θ − β4 y sin θ
λ
2 − 2a
satisfies the derivative decay estimates
v −2 |∂θ φ|+v −1 |∂y ∂θ φ|+v −2 |∂θ2 φ|+v −1 |∂y2 ∂θ φ|+v −2 |∂y ∂θ2 φ|+v −3 |∂θ3 φ| . b(t)2−ε1 ,
and the C 0 decay estimates
|φ(y, θ, t)|
3
(1 + y 2 ) 2
. b(t)2−ε2
and
|φ(y, θ, t)|
11
(1 + y 2 ) 20
. b(t)1−ε1 ,
all holding uniformly as tn % T .
Acknowledgment. The authors are deeply grateful to I.M. Sigal for many helpful
discussions related to extensions of their work in [13].
2. Preliminaries
2.1. Notation. We study the evolution of embedded graphs over a cylinder S1 × R
in R3 . We consider initial data M0 expressed in a cylindrical coordinate system by
a positive function u0 (x, θ). Then for as long as the flow remains a graph, all Mt
are determined by r = u(x, θ, t), where, as we showed in [13], u satisfies the initial
condition u(x, θ, 0) = u0 (x, θ) and evolves by
(2.1)
∂t u =
{1 + ( ∂uθ u )2 }∂x2 u +
2
1+(∂x u)2 2
θ u)
∂θ u − 2 (∂x u)(∂
∂x ∂θ u
u2
u3
∂θ u 2
2
1 + (∂x u) + ( u )
−
(∂θ u)2
u3
−
1
.
u
UNIVERSALITY IN MCF NECKPINCHES
5
As in [13], we apply adaptive rescaling, transforming the original space-time
variables x and t into rescaled blowup variables2
Z t
λ−2 (s) ds,
y(x, t) := λ−1 (t)x
and
τ (t) :=
0
where λ > 0 is a scaling parameter to be chosen, with λ(0) = λ0 . The equation’s
scaling symmetry allows us, without loss of generality, to fix λ0 = 1. Note that
y = 0 is only the approximate center of the developing neckpinch; we address this
issue below. We define a rescaled radius v(y, θ, τ ) by
(2.2)
v y(x, t), θ, τ (t) := λ−1 (t) u(x, θ, t).
Then v initially satisfies v(y, θ, 0) = v0 (y, θ), where v0 (y, θ) := λ−1
0 u0 (λ0 y, θ).
With respect to commuting (y, θ, τ ) derivatives, the quantity v evolves by
∂τ v = Av v + av − v −1 ,
(2.3)
where Av is the quasilinear elliptic operator
(2.4) Av := F1 (p, q)∂y2 + v −2 F2 (p, q)∂θ2 + v −1 F3 (p, q)∂y ∂θ + v −2 F4 (p, q)∂θ − ay∂y .
The coefficients of Av are defined by
1 + q2
,
1 + p2 + q 2
F1 (p, q)
:=
F3 (p, q)
:= −
F2 (p, q)
:=
1 + p2
,
1 + p2 + q 2
F4 (p, q)
:=
q
,
1 + p2 + q 2
(2.5)
2pq
,
1 + p2 + q 2
where
(2.6)
a := −λ∂t λ,
p := ∂y v,
and
q := v −1 ∂θ v.
Before stating our assumptions, we require some further notation. We denote
the formal solution of the adiabatic approximation to equation (2.3) by
r
2 + sy 2
(2.7)
Vr,s (y) :=
,
2 − 2r
where r and s are positive parameters. We introduce a step function
 19 √
 20 2 if sy 2 < 20,
(2.8)
g(y, s) :=

4
if sy 2 ≥ 20.
This function differs from that used in [13] in that we here prescribe a slightly
sharper constant in the inner region {βy 2 ≤ 20}, where
−1
β(τ ) := κ0 + τ
.
This modification is not essential but simplifies some of the extra work required to
compensate for the lack of discrete symmetry assumptions in this paper.3
2In the sequel, we abuse notation by using whichever time scale (t or τ ) is most convenient.
3Here, κ 1 is a large constant to be fixed below. For simplicity, we may without loss of
0
generality set κ0 ≡ b−1
0 , where b0 1 is the constant introduced in Section 2.2.
6
ZHOU GANG AND DAN KNOPF
We introduce a Hilbert space L2µ ≡ L2 (S1 × R; dθ µ dy), with weighted measure
3
µ(y) := (M + y 2 )− 5 defined with respect to a constant M 1 to be fixed below.4
We denote the L2µ inner product by
Z Z
hϕ, ψiµ :=
ϕψ dθ µ dy,
S1
R
and its norm by k · kµ . The inner product in the complex Hilbert space L2 (S1 ) is
denoted by
Z
ϕψ dθ,
hϕ, ψiS1 :=
S1
and its norm is k · kS1 . The undecorated inner product h·, ·i denotes the standard
(unweighted) inner product in L2 (S1 × R).
We define a norm k · km,n by
m
kφkm,n := (1 + y 2 )− 2 ∂yn φ ∞ .
L
We write ϕ .p
ψ if there exists a uniform constant C > 0 such that ϕ ≤ Cψ, and
we set hxi := 1 + |x|2 . Finally, for any f (y, θ, τ ), we define
1
hf (y, θ, τ ), e±iθ iS1 .
(2.9)
f± (y, τ ) :=
2π
2.2. Main Assumptions. There are small positive constants b0 , c0 such that:
[A1] The initial surface is a graph over S1 × R determined by a smooth function
u0 (x, θ) > 0. The function u0 is uniformly (2b0 )-Lipschitz and satisfies
u0 (x, ·) ≥ c∗ Va0 ,b0 (x) for some c∗ > 0, along with the estimates 5
|(u0 )± | + |(∂x u0 )± | < b20 ,
21/10
,
53/20
.
khxi−5 ∂θ u0 kL∞ < b0
khxi−11/10 (u0 )± kL∞ + khxi−11/10 (∂x u0 )± kL∞ < b0
[A2] The initial function satisfies u0 (x, ·) > g(x, b0 ).
[A3] The initial surface is a small deformation of a formal solution Va0 ,b0 (x) in
the sense that for (m, n) ∈ {(3, 0), (11/10, 0), (2, 1), (1, 1)}, one has
m+n
1
2 + 10
ku0 (·) − Va0 ,b0 (·)km,n < b0
.
[A4] The parameter a0 = a(0) obeys the bound |a0 − 1/2| < c0 .
[A5] The initial surface obeys the further pointwise derivative bounds
P
−n m n
2
n6=0, 2≤m+n≤3 u0 |∂x ∂θ u0 | < b0 ,
−1/2
b0 u0
1/2
3
|∂x u0 | + b0 |∂x2 u0 | + |∂x3 u0 | < b02 ,
3
|∂x ∂θ2 u0 | + u−1
0 |∂θ u0 | < c0 .
[A6] The initial surface obeys the Sobolev bounds
X
4/5
m n
4
b0 k∂x4 u0 kµ + k∂x5 u0 kµ +
ku−n
0 ∂x ∂θ u0 kµ < b0 .
n6=0, 4≤m+n≤5
m n
[A7] The initial surface satisfies ku−n
0 ∂x ∂θ u0 kL∞ < ∞ for 4 ≤ m + n ≤ 6.
4In [13], µ and M were denoted by σ and Σ, respectively.
5Smallness of (u ) and (∂ u ) in the inner region suffice for the first bootstrap machine,
x 0 ±
0 ±
but global smallness is needed for certain propagator estimates in the second bootstrap machine.
UNIVERSALITY IN MCF NECKPINCHES
7
Three remarks are in order here: (i) Our choice λ0 = 1 guarantees that x = y
and τ = 0 both hold at t = 0, hence that the assumptions above apply identically
to u(x, θ, 0) = u0 (x, θ) and v(y, θ, 0) = v0 (y, θ). (ii) The lower bounds for u0 in
Assumptions [A1] and [A2] are clearly related, with [A1] being stronger for large
|x|, and [A2] being stronger for small |x|. The bounds are presented in this way
so that [A1] is the only member of [A1]–[A6] that is significantly changed from our
earlier work [13], other than the slight modification of the function g used in [A2].
The new inequalities in [A1] compensate for the removal of the discrete symmetries
that were used in [13]; we explain this further below. (iii) Assumption [A7], which
is also new, imposes slightly more control at spatial infinity than we needed in [13].
This allows us to integrate by parts in y without needing the hypothesis of reflection
symmetry that we used in that earlier paper. (See Remark 2 below.)
2.3. Strategy of the proof. In earlier work [13] (also see [14]) we decomposed
a solution v(y, θ, τ ) of equation (2.3) into two terms: a point Va(τ ),b(τ ) (y) on a
manifold M of approximate solutions, and a remainder term φ(y, θ, τ ) := v(y, θ, τ )−
Va(τ ),b(τ ) (y) approximately orthogonal to that manifold. These terms represent the
large, slowly-changing part of the solution, and the small, rapidly-decaying part,
respectively. The utility of this decomposition is that permutations of v tangent
to M are linearly unstable, whereas those orthogonal to M are linearly stable.
We follow the same strategy here, except that our lack of symmetry assumptions
requires us to construct a more refined decomposition in order to compensate for
coordinate instabilities of (2.3). The details of this refinement are as follows.
In [13], we imposed discrete symmetries on the initial surface: u0 (−x, ·) = u0 (x, ·)
and u0 (·, θ + π) = u0 (·, θ). These assumptions fixed the cylindrical axis and center
y = 0 of the developing neckpinch singularity. In the present paper, we remove these
assumptions, which means that we must determine the unique cylinder and center
at which the neckpinch forms. To compensate, Assumption [A1] is strengthened
from [13]. Its revised form ensures that a neck aligned to a slightly perturbed
cylinder can still be written as a normal graph, and provides extra estimates (to be
improved in the second bootstrap machine) for the θ-dependence of the solution.
These estimates effectively replace the lost symmetries, which implied inequalities
(like k∂θ2 vkS1 ≥ 2k∂θ vkS1 ) that we cannot use here.
We determine the unique cylinder and center of the neckpinch as follows. First, at
some fixed time τ1 , we construct “optimal coordinates” for the developing solution.
As motivation, we observe that linearizing equation (2.3) at the cylinder of radius
a−1/2 leads to the elliptic operator
La = Ha − a∆ ≡ −(∂y2 + ay∂y + 2a) − a∆,
where ∆ is the Laplacian of S1 . The operator −Ha = ∂y2 + ay∂y + 2a is commonly
seen in geometric evolution equations. It is self-adjoint in the weighted Hilbert space
2
L2 (R, e−ay /2 dy) with spectrum spec(Ha ) = {a(j − 2)}j≥0 . Its eigenfunctions are
the Hermite polynomials ha,j ; in particular, its (weakly) unstable eigenspaces are
spanned by ha,0 = 1, ha,1 = y, and ha,2 = y 2 − a−1 . The spectrum {k 2 }k≥0
of −∆ is also well known, and its lowest nontrivial
eigenspace is
spanned by the
restriction to S1 of the coordinate functions x1 S1 = cos θ and x2 S1 = sin θ. Hence
spec(La ) = {a(j − 2) + ak 2 ; j ≥ 0, k ≥ 0}, and we can summarize the (weakly)
8
ZHOU GANG AND DAN KNOPF
unstable eigenvalues of La and their associated eigenfunctions in this table:
Eigenvalue
−2a
−a
0
Multiplicity
1
3
3
Eigenfunction(s)
1
y = λ−1 x0 , x1 , x2
y 2 − a−1 ; yx1 , yx2
Geometric meaning
rescale cylinder
translate center of neck
shape neck; tilt cylinder
In optimal coordinates, a surface is orthogonal to the span of these functions.
Definition 1. We say a cylindrical coordinate system (x, r, θ) is an optimal system
at time τ1 with respect to positive parameters λopt ≡ λopt (τ1 ) and bopt ≡ bopt (τ1 ) if
s
2 + bopt (τ1 )y 2
φopt (y, θ, τ1 ) := v(y, θ, τ1 ) − Vbopt (τ1 )/2, bopt (τ1 ) ≡ v(y, θ, τ1 ) −
1 + 12 bopt (τ1 )
satisfies the orthogonality condition
φopt ⊥ {1, y, y 2 − a−1
1 , cos θ, sin θ, y cos θ, y sin θ}
in L2 (R × S1 , dθ e−a1 y
2
/2
−1
dy), where v = λ−1
opt u, y = λopt x, and a1 :=
1
2
− 14 bopt .
Having constructed optimal coordinates at time τ1 , we then observe that by a
straightforward extension of the implicit function theorem argument developed in
[14], there exists a (possibly small) earlier time interval [τ0 , τ1 ] and C 1 functions
λ(τ ), a(τ ), b(τ ), and β0 (τ ), . . . , β4 (τ ) defined for τ ∈ [τ0 , τ1 ], such that the quantity
s
2 + b(τ )y 2
(2.10)
φ̃(y, θ, τ ) := v(y, θ, τ ) − V3/2−2a(τ ),b(τ ) ≡ v(y, θ, τ ) −
2 − 2a(τ )
can be parameterized in the form
(2.11)
φ̃(y, θ, τ ) = β0 y + β1 cos θ + β2 sin θ + β3 y cos θ + β4 y sin θ + φ(y, θ, τ ),
where the orthogonality conditions
φ ⊥ {1, y, y 2 − a−1 , cos θ, sin θ, y cos θ, y sin θ}
2
hold in L2 (R × S1 , dθ e−a(τ )y /2 dy) for τ ∈ [τ0 , τ1 ], with the boundary conditions
λ(τ1 ) = λopt (τ1 ), a(τ1 ) = a1 (recall that a = −λ∂t λ), and b(τ1 ) = bopt (τ1 ), and
with the new “translate” and “tilt” parameters satisfying the boundary conditions
(2.12)
β0 (τ1 ) = β1 (τ1 ) = β2 (τ1 ) = β3 (τ1 ) = β4 (τ1 ) = 0.
The next step of the argument, accomplished in the second bootstrap machine,
is to obtain sufficiently good control on the parameters β0 , . . . , β4 to conclude that
the sequence of optimal coordinate systems constructed at discrete times τ (t1 ) <
τ (t2 ) < · · · < T converges. We accomplish this by proving that the difference
between an optimal system at time t1 and an optimal system at a later time t2 is
of order b2 (t1 ), where b(t) = {1 + o(1)}(− log(T − t))−1 → 0 as t % T .
This concludes our heuristic outline of the proof. We now provide the details
needed to make everything rigorous.
UNIVERSALITY IN MCF NECKPINCHES
9
3. How the solution evolves
Given integers m, n ≥ 0 and a real number k ≥ 0, we define
vm,n,k := v −k (∂ym ∂θn v).
(3.1)
The following result is an immediate consequence of Lemma 3.1 and Corollary 3.3
from our earlier work in [13]:
2
Lemma 1. The quantity vm,n,k
evolves by
2
2
2
∂τ (vm,n,k
) = Av (vm,n,k
) + 2 (k + 1)v −2 − (m + k − 1)a vm,n,k
(3.2)
−Bm,n,k + 2Em,n,k vm,n,k ,
2
where Bm,n,k := Av (vm,n,k
) − 2vm,n,k Av vm,n,k satisfies the inequality
(∂y vm,n,k )2 + (v −1 ∂θ vm,n,k )2
≥ 0,
1 + p2 + q 2
P5
and where the commutator terms Em,n,k = `=0 Em,n,k,` are given by
(3.3)
Bm,n,k ≥ 2
Em,n,k,0
:= −kv −1 vm,n,k (Av v + ay∂y v),
Em,n,k,1
:= v −k ∂ym ∂θn (F1 ∂y2 v) − F1 ∂y2 vm,n,k ,
Em,n,k,2
:= v −k ∂ym ∂θn (v −2 F2 ∂θ2 v) − v −2 F2 ∂θ2 vm,n,k ,
Em,n,k,3
:= v −k ∂ym ∂θn (v −1 F3 ∂y ∂θ v) − v −1 F3 ∂y ∂θ vm,n,k ,
Em,n,k,4
:= v −k ∂ym ∂θn (v −2 F4 ∂θ v) − v −2 F4 ∂θ vm,n,k ,
Em,n,k,5
:= −v −k ∂ym ∂θn (v −1 ) − v −2 vm,n,k .
(3.4)
The utility of estimate (3.3) is that once one has suitable first-order estimates for
v, one can bound 1+p2 +q 2 from above, whereupon Bm,n,k contributes useful higher2
2
order terms of the form −ε(vm+1,n,k
+ vm,n+1,k+1
) to the evolution equation (3.2)
2
satisfied by vm,n,k
.
In the bootstrapping arguments made in this paper, we must estimate the nonlinear “error terms” Em,n,k,` defined in display (3.4). In Section 3 of [13], we made
the following observations, which we freely use here in estimating these quantities.
Remark 1. For any i, j ≥ 0 and ` = 1, . . . , 4, there exist constants Ci,j,` such that
|∂pi ∂qj F` (p, q)| ≤ Ci,j,`
for all p, q ∈ R. Moreover, Em,n,k,0 = 0 if k = 0, and Em,n,k,5 = 0 if m + n = 1.
4. The first bootstrap machine
4.1. Inputs. We now present the inputs to our first bootstrap machine, whose
structure we describe below. By standard regularity theory for quasilinear parabolic equations, if the initial data satisfy the Main Assumptions in Section 2.2 for
10
ZHOU GANG AND DAN KNOPF
sufficiently small b0 and c0 , then solutions originating from such data will satisfy
the properties below for a sufficiently short time interval [0, τ1 ].6
Most of these properties are global, while others are local in nature. In many
of the arguments that follow, we separately treat the inner region {βy 2 ≤ 20} and
the outer region {βy 2 ≥ 20}, where β(τ ) := (κ0 + τ )−1 . Note that the inner region
here corresponds to the union of the parabolic and intermediate regions in [3]. Note
also that our Main Assumptions imply that slightly stronger conditions hold for
the inner region than for the outer. This is natural, because it is in the inner region
that one expects the solution v of (2.3) to be closest to the formal solution V1/2,β of
the equation 21 y∂y V − 12 V + V −1 = 0 that is the adiabatic approximation of (2.3).
To state the global conditions, we decompose the solution into θ-independent
and θ-dependent parts v1 , v2 , respectively, defined by
Z 2π
1
v(y, θ, τ ) dθ and v2 (y, θ, τ ) := v(y, θ, τ ) − v1 (y, τ ).
(4.1) v1 (y, τ ) :=
2π 0
Here are the global conditions:
[C0] For τ ∈ [0, τ1 ], the solution has a uniform lower bound v(·, ·, τ ) ≥ κ−1
0 .
[C1] For τ ∈ [0, τ1 ], the solution satisfies the first-order estimates
2
1
|∂y v| . β 5 v 2 ,
3
|∂θ v| . β 2 v 2 ,
and |∂θ v| . v.
[C2] For τ ∈ [0, τ1 ], the solution satisfies the second-order estimates
3
|∂y2 v| . β 5 ,
3
|∂y ∂θ v| . β 2 v,
|∂y ∂θ v| . 1,
3
and |∂θ2 v| . β 2 v 2 .
[C3] For τ ∈ [0, τ1 ] the solution satisfies the third-order decay estimates
|∂y3 v| . β
3
and v −n |∂ym ∂θn v| . β 2
for m + n = 3 with n ≥ 1, as well as a “smallness estimate” that
1
11
β − 20 |∂y3 v| + |∂y2 ∂θ v| + |∂y ∂θ2 v| + v −1 |∂θ3 v| . (β0 + ε0 ) 40
holds for some ε0 = ε0 (b0 , c0 ) 1.7
[Ca] For τ ∈ [0, τ1 ], the parameter a satisfies
1
1
− κ−1
+ κ−1
0 ≤a≤
0 .
2
2
1
[Cg] For τ ∈ [0, τ1 ], one has hyi−1 |∂y v| ≤ M 4 β, where M 1 is the constant
introduced in Section 2.1.
[Cr] There exists 0 < δ 1 such that the scale-invariant bound |v2 | ≤ δv1 holds
everywhere for τ ∈ [0, τ1 ].
[Cs] For τ ∈ [0, τ1 ], one has Sobolev bounds kv −n ∂ym ∂θn vkµ < ∞ whenever
4 ≤ m + n ≤ 7.
[Ct] For τ ∈ [0, τ1 ], the solution is “tame at infinity” in the sense that for
4 ≤ m + n ≤ 7, one has kv −n ∂ym ∂θn vkL∞ < ∞.
6In order to obtain some of the derivative bounds here, we use a general interpolation result,
Lemma B.2 from [13]. Note that Lemmas B.1 and B.2 in [13] use only the 2π-periodicity of
v(·, θ, ·), hence apply here as well.
7The smallness estimate is only used in the proof of Theorem 4.
UNIVERSALITY IN MCF NECKPINCHES
11
Note that the global
p gradient Condition [Cg] posits the β1decay rate of the formal
solution V (y, τ ) = 2 + βy 2 , but with a large constant M 4 1. The effect of our
bootstrap argument will be to sharpen that constant. Remark 3 (below) shows that
Condition [Cb] in Section 6, which is directly implied by our Main Assumptions in
Section 2.2, in turn implies that the global gradient bound [Cg] holds, along with
extra properties that are local to the inner region.
The extra conditions for the inner region are as follows.
[C0i] For τ ∈ [0, τ1 ] and βy 2 ≤ 20, the quantity v is uniformly bounded from
above and below: there exists C∗ such that
19 √
2 ≡ g(y, β) ≤ v(y, ·, ·) ≤ C∗ .
20
[C1i] For τ ∈ [0, τ1 ] and βy 2 ≤ 20, the solution satisfies the stronger first-order
estimates
1
1
|∂y v| . β 2 v 2
−1
and |∂θ v| . κ0 2 v.
[C2i] For τ ∈ [0, τ1 ] and βy 2 ≤ 20, the θ-dependent part 8 of the solution satisfies
|v± | + |(∂y v)± | . β 2 .
The only differences above from [13] are Condition [Ct], which follows from the
new Assumption [A7], and [C2i], which follows from the revised [A1]. As explained
in Section 2.3 and Remark 2 (below), [Ct] and [C2i] allow us to compensate for
the discrete symmetries assumed in [13] that may not hold here. We make the
consequences of these changes clear in the proofs that follow.
4.2. Outputs. The output of this machine consist of the following estimates, which
collectively improve Conditions [C0]–[C3], [Cs], [Cr], [Cg], and [C0i]–[C1i]:
v(y, θ, τ ) ≥ g(y, β),
1
1
v − 2 |∂y v| . β 2 ,
|∂y v| ≤ ε0 ,
v = O(hyi) as |y| → ∞;
−1
33
v −2 |∂θ v| . β 20 ,
v −1 |∂θ v| . κ0 2 ;
33
β|∂y2 v| + v −1 |∂y ∂θ v| + v −2 |∂θ2 v| . β 20 ;
1
|∂y ∂θ v| + v −1 |∂θ2 v| . (β0 + ε0 ) 20 ;
1
33
β 2 |∂y3 v| + v −1 |∂y2 ∂θ v| + v −2 |∂y ∂θ2 v| + v −3 |∂θ3 v| . β 20 ;
11
1
β − 20 |∂y3 v| + |∂y2 ∂θ v| + |∂y ∂θ2 v| + v −1 |∂θ3 v| . (β0 + ε0 ) 20 .
Here, β0 ≡ β(0) and ε0 are independent of τ1 . Therefore, these improvements allow
us to propagate the assumptions above forward in time in our proof of the Main
Theorem.
8Recall that v and (∂ v) are defined by (2.9).
y ±
±
12
ZHOU GANG AND DAN KNOPF
4.3. Structure. In the first step of the bootstrap argument, we derive improved
estimates for v and its first derivatives in the outer region.
Theorem 1. Suppose a solution v = v(y, θ, τ ) of (2.3) satisfies Assumption [A1]
at τ = 0. If for τ ∈ [0, τ1 ], Conditions [Ca] and [C0]–[C2] hold in the outer region
{βy 2 ≥ 20}, and Conditions [C0i]–[C1i] hold on its boundary, then throughout the
outer region {βy 2 ≥ 20}, one has estimates
(4.2)
v(y, ·, ·) ≥ 4,
(4.3)
|∂y v| . β 2 v 2 ,
(4.4)
|∂y v| ≤ ε0 ,
(4.5)
|∂θ v| ≤ C0 κ0 2 v,
1
1
−1
for all τ ∈ [0, τ1 ], where ε0 and C0 depend only on the initial data and not on τ1 .
Proof. Except for estimate (4.4), the statements and their proofs are identical to
those stated in Theorem 4.2 and proved in Section 5 of [13]. Indeed, the reader
may check that the proofs there used neither the fact that v(y, ·, ·) was assumed
to be an even function of y, nor that v(y, ·, ·) was assumed to be orthogonal to the
lowest two eigenspaces span{1, sin θ, cos θ} ⊂ L2 ({y} × S1 ) of the Laplacian on S1 .
We now show how to improve the bound |∂y v| . 1 proved in in Section 5 of [13]
to the bound |∂y v| ≤ ε0 needed here. In our earlier work, we showed that 1 + (∂y v)4
was bounded above for all time by the solution of the ivp
6
d
ϕ = Cβ 5 ϕ,
dτ
ϕ(0) = sup{1 + (∂y v)4 }.
τ =0
Noting that
limτ →∞ ϕ(τ )
= exp
ϕ(0)
5C
!
1/5
,
κ0
where C does not increase as we increase κ0 , we obtain the result by choosing κ0
sufficiently large and b0 , hence ϕ(0), sufficiently close to 1.
Note that estimate (4.4) provides ε0 such that |∂y v| ≤ ε0 in the outer region for
√
1
τ ∈ [0, τ1 ]. By Condition [C0i], one has v(± 20β − 2 ) ≤ c2 for some c > 0. Hence
by quadrature, one obtains the immediate corollary that
(4.6)
v = O(hyi)
as
|y| → ∞.
In the second step, we derive estimates for second and third derivatives of v. In
the outer region, these estimates are obtained by maximum principle arguments,
exactly as in [13], using the noncompact maximum principle derived in Appendix C
of that paper. In the inner region, we introduce and bound suitable Lyapunov functionals and then employ Sobolev embedding. These arguments require integration
by parts.
UNIVERSALITY IN MCF NECKPINCHES
13
Remark 2. For 1 ≤ m + n ≤ 7, Condition [Cs] implies that kvm,n,n kL∞ < ∞
for τ ∈ [0, τ1 ]. Using this and (4.6), it is straightforward to verify that we may
integrate by parts in y as was done in [13] — without using reflection symmetry.
Two special but important examples are the facts that for 0 ≤ r + s, m + n ≤ 5, one
has
h∂y2 vm,n,n , vr,s,s iµ = −hvm+1,n,n , vr+1,s,s + vr,s,s ∂y (log µ)iµ ;
and for 1 ≤ m + n ≤ 5, one has
1
1
kvm,n,n k2µ + hvm,n,n , y∂y (log µ)vm,n,n iµ
2
2
1
≤ kvm,n,n k2µ ,
2
−hy∂y vm,n,n , vm,n,n iµ =
because y∂y (log µ) ≤ 0.
Theorem 2. Suppose that a solution v = v(y, θ, τ ) of equation (2.3) satisfies Assumption [A1] at τ = 0, and Conditions [Ca], [C0]–[C3], [Cs], [Cr], [Cg], and
[C0i]–[C2i] for τ ∈ [0, τ1 ]. Then for the same time interval, the solution satisfies
the following pointwise bounds throughout the inner region {βy 2 ≤ 20}:
33
β|∂y2 v| + v −1 |∂y ∂θ v| + v −2 |∂θ2 v| . β 20
(4.7)
and
(4.8)
1
33
β 2 |∂y3 v| + v −1 |∂y2 ∂θ v| + v −2 |∂y ∂θ2 v| + v −3 |∂θ3 v| . β 20 ,
The fact that we do not achieve the expected β 2 decay on the rhs of estimates (4.7)–(4.8) is due to our use of Sobolev embedding with respect to the
weighted measure for L2µ introduced in Section 2.1.
We prove Theorem 2 in Section 5. This task requires us to modify the arguments
used in [13]: those arguments use evenness of v(y, ·, ·) and π-periodicity of v(·, θ, ·),
neither of which are assumed here.
Using Theorem 2 to ensure that they hold on the boundary of the inner region,
we then extend its estimates to the outer region.
Theorem 3. Suppose that a solution v = v(y, θ, τ ) of equation (2.3) satisfies Assumption [A1] at τ = 0, and Conditions [Ca] and [C0]–[C3] for βy 2 ≥ 20 and
τ ∈ [0, τ1 ]. Then the estimates
33
β|∂y2 v| + v −1 |∂y ∂θ v| + v −2 |∂θ2 v| . β 20
(4.9)
and
(4.10)
1
33
β 2 |∂y3 v| + v −1 |∂y2 ∂θ v| + v −2 |∂y ∂θ2 v| + v −3 |∂θ3 v| . β 20
hold throughout the outer region {βy 2 ≥ 20} for the same time interval, provided
that they hold on the boundary {βy 2 = 20}.
As an easy corollary, we apply general interpolation results (c.f. [13, Lemma B.2])
to get our final first-order estimate claimed in Section 4.2, namely
|∂θ v|
1
|∂θ v2 |
C1
|∂θ2 v2 |
≤
≤
max
2
2
2
2
v
(1 − δ) v1
(1 − δ) θ∈[0,2π] v12
33
(1 + δ)2
|∂θ2 v|
≤ C2
max
≤ C3 β 20 .
(1 − δ)2 θ∈[0,2π] v 2
14
ZHOU GANG AND DAN KNOPF
The proof of Theorem 3 is identical to that of Theorem 4.6 in Sections 7.1–7.2
of [13], where neither evenness in y nor π-periodicity in θ are used.
In the final step, we improve the “smallness estimates” in Condition [C3], producing improved bounds for |∂y ∂θ2 v| and v −1 |∂θ3 v| that serve as inputs to the second
bootstrap machine.
Theorem 4. Suppose that a solution v = v(y, θ, τ ) of equation (2.3) satisfies Assumption [A1] at τ = 0, and Conditions [Ca], [C0]–[C3], [Cs], [Cr], [Cg], and
[C0i]–[C1i] for τ ∈ [0, τ1 ]. Then for the same time interval, the solution satisfies
11 1
β − 10 (∂y3 v)2 + (∂y2 ∂θ v)2 + (∂y ∂θ2 v)2 + v −2 (∂θ3 v)2 . (β0 + ε0 ) 10 .
As an easy corollary, we again apply general interpolation results to get our final
second-order estimates claimed in Section 4.2, namely
1
|∂y ∂θ v| + v −1 |∂θ2 v| . (β0 + ε0 ) 20 .
Once Theorems 2 and 3 are established, the proof of Theorem 4 is identical to
that of Theorem 4.7 in Section 7.3 of [13].
Collectively, Theorems 1–4 complete the first bootstrap machine.
5. Improved estimates for the inner region
In this section, we prove Theorem 2.
Given integers m, n ≥ 0, we define Lyapunov functionals Ωm,n = Ωm,n (τ ) by
Z ∞ Z 2π
(5.1)
Ωm,n := kvm,n,n k2µ ≡
v −2n (∂ym ∂θn v)2 dθ µ dy.
−∞
0
Note that vm,n,k is defined in equation (3.1), and µ = µ(y) is defined in Section 2.1.
By Conditions [C2], [C3], and [Cs], these functionals and their τ -derivatives are
well defined if m + n ≤ 5.
We prove the second- and third-order derivative bounds in Theorem 2 in two
steps. (i) We bound certain weighted sums of Ωm,n with 2 ≤ m + n ≤ 5. Here, as
in [13], it suffices to show that derivatives of orders three through five decay at the
same rates as the estimates we derive for those of second order, rather than at the
faster rates one would expect from parabolic smoothing. This somewhat reduces
the work involved. (ii) We apply Sobolev embedding to get pointwise estimates,
6
1
using the facts that |y| . β − 2 in the inner region, and that µ ∼ |y|− 5 as |y| → ∞.
Our lack of discrete symmetry assumptions, in contrast to [13], requires us to
modify some of the proofs from our earlier work. Here, we employ three kinds
of argument: one for functionals with only y-derivatives, one for functionals with
“many” y-derivatives, and one for functionals with “few” y-derivatives. These categories are made precise below.
As immediate consequences of the proofs of Lemmas 6.1–6.3 in [13], we find that
the following differential inequalities for the second-order functionals hold pointwise:
Lemma 2. There exists 0 < C < ∞ such that:
2
(i) the quantity (v2,0,0 ) = (∂y2 v)2 satisfies
1
2
2
∂τ (v2,0,0 ) ≤(v2,0,0 )Av (v2,0,0 ) + (v −2 − a)(v2,0,0 )
2
11 + C β|(v2,0,0 )| + β 10 |(v3,0,0 )| + |(v2,1,1 )| + |(v1,2,2 )| ;
UNIVERSALITY IN MCF NECKPINCHES
15
2
(ii) the quantity (v1,1,1 ) = v −2 (∂y ∂θ v)2 satisfies
1
2
2
∂τ (v1,1,1 ) ≤(v1,1,1 )Av (v1,1,1 ) + (2v −2 − a)(v1,1,1 )
2
+ C β 2 |(v1,1,1 )| + |(v2,1,1 )| + |(v1,2,2 )| + |(v0,3,3 )| + β 3 |(v3,0,0 )| ;
(iii) the quantity (v0,2,2 )2 = v −4 (∂θ2 v)2 satisfies
1
2
2
∂τ (v0,2,2 ) ≤(v0,2,2 )Av (v0,2,2 ) + (3v −2 − a)(v0,2,2 )
2
21
+ C β 2 |(v1,2,2 )| + |(v0,3,3 )| + β 10 |(v0,2,2 )| + β 3 |(v2,1,1 )| .
We also import various pointwise estimates for derivatives of the coefficients
F` of the quasilinear elliptic operator Av defined in (2.4)–(2.5). They appear as
estimates (6.2)–(6.6) in [13], and follow from Theorem 1, Condition [C2], and Conditions [C0i] and [C1i] for the inner region. We state them here as:
Lemma 3. First-order derivatives of the coefficients F` obey the bounds
3
|∂y F` | . β 5
and
3
v −1 |∂θ F` | . β 2 .
Second-order derivatives of the coefficients F` obey the bounds
6
|∂y2 F` | . β 5 + |(v3,0,0 )| + |(v2,1,1 )|,
v −1 |∂y ∂θ F` | . β 2 + |(v2,1,1 )| + |(v1,2,2 )|,
v −2 |∂θ2 F` | . β 3 + |(v1,2,2 )| + |v3,3,0 |.
In practice, we use these estimates in the following forms, that follow immediately from combining Lemma 3 with Conditions [C1]–[C3], along with the fact that
kµ−1 ∂y2 µkL∞ = O(M −1 ) as M → ∞. The statement is:
Lemma 4. One has the pointwise estimates
6
1
3
µ−1 |∂y2 (µF1 )| . |(v3,0,0 )| + |(v2,1,1 )| + β 5 + M − 2 β 5 + M −1 . β + M −1 ,
3
3
|∂θ2 (v −2 F2 )| . |v3,3,0 | + |(v1,2,2 )| + β 2 . β 2 ,
1
3
1
3
µ−1 |∂y ∂θ (v −1 µF3 )| . |(v2,1,1 )| + |(v1,2,2 )| + (1 + M − 2 )β 2 . (1 + M − 2 )β 2 ,
3
|∂θ (v −2 F4 )| . β 2 .
5.1. Estimating the time evolution of the Lyapunov functionals. In this
section, we prove estimates for the evolution equations satisfied by Ωm,n . The
proofs and results are similar but not identical to those in Lemmas 6.5–6.9 and
6.11–6.12 of [13]. The arguments here come in three flavors: one for the case that
there are no θ derivatives (Ωm,0 with 2 ≤ m ≤ 5), another for the case that there are
“many” θ derivatives (Ωm,n with m ∈ {0, 1} and m + n ∈ {3, 4, 5}), and a third for
the remaining cases ((Ωm,n with (m, n) ∈ {(2, 1), (3, 1), (2, 2), (4, 1), (3, 2), (2, 3)}).
16
ZHOU GANG AND DAN KNOPF
Lemma 5. There exist constants 0 < ε < C < ∞ and ρ = ρ(M ) > 0, with
ρ(M ) & 0 as M → ∞, such that
1
o
n 1
1
1
1
11
d
2
2
2
2
+ β 2 Ω3,0
+ β 10 Ω2,1
+ Ω1,2
,
Ω2,0 ≤ −ε (Ω2,0 + Ω3,0 + Ω2,1 ) + ρ βΩ2,0
dτ
o
n 1
1
1
1
1
3
d
2
2
2
2
+ β 10 Ω4,0
+ Ω3,1
+ Ω2,2
Ω3,0 ≤ −ε (Ω3,0 + Ω4,0 + Ω3,1 ) + ρβ 2 Ω3,0
dτ
1
1
1
2
2
+ Cβ {Ω3,0 + Ω2,1 + Ω1,2 + Ω0,3 } + Cβ 2 Ω3,0
Ω2,0
,
1
d
Ω4,0 ≤ −ε (Ω4,0 + Ω5,0 + Ω4,1 ) + Cβ 2
dτ
1
d
Ω5,0 ≤ −ε (Ω5,0 + Ω6,0 + Ω5,1 ) + Cβ 2
dτ
X
8
Ωi,j + Cβ 5
4≤i+j≤5
X
X
1
2
,
Ωi,j
4≤i+j≤5
21
Ωi,j + Cβ 10
5≤i+j≤6
X
1
2
Ωi,j
.
5≤i+j≤6
Proof. We provide a detailed argument for Ω2,0 , and sketch the remaining cases.
Argument for Ω2,0 : Applying Lemma 2 and Cauchy–Schwarz in L2µ , we get
1 d
Ω2,0 ≤h(v2,0,0 ), Av (v2,0,0 )iµ + h(v −2 − a)(v2,0,0 ), (v2,0,0 )iµ
2 dτ
n 1
o
1
1
1 11
2
2
2
2
+ ρ βΩ2,0
+ β 10 Ω3,0
+ Ω2,1
+ Ω1,2
.
Define I1 := h(v2,0,0 ), Av (v2,0,0 )iµ and I2 := h(v −2 − a)(v2,0,0 ), (v2,0,0 )iµ . Here ρ is
a constant that can be made as small as desired, because
Z
∞
Z
ρ=
2π
Z
∞
dθ µ(y) dy = 2π
−∞
M + y2
− 53
dy,
−∞
0
which tends to zero as M % ∞.9
By Remark 2, we may integrate I2 by parts in y, obtaining
I2 = −aΩ2,0 + h(v2,0,2 ), (v2,0,0 )iµ
≤ −aΩ2,0 + h(v1,0,2 ), 2(v2,0,0 )(v1,0,1 ) − (v3,0,0 ) − (v2,0,0 )∂y (log µ)iµ
1
1 1
2
2
+ β 2 Ω3,0
.
≤ −aΩ2,0 + ρ βΩ2,0
1
In the last step, we used the facts that k∂y log µkL∞ = O(M − 2 ) as M → ∞ and
that ∂y (log µ) = O(|y|−1 ) as |y| → ∞, along with Condition [Cg] and estimate (4.3).
Again using Remark 2 and integrating by parts, we find that
a
I1 ≤ I∗ + Ω2,0 ,
2
9Recall that µ(y) := (M + y 2 )− 53 , where M is an arbitrarily large constant.
UNIVERSALITY IN MCF NECKPINCHES
17
where I∗ := h(v2,0,0 ), F1 (v4,0,0 )+F2 (v2,2,2 )+F3 (v3,1,1 )+F4 (v2,1,2 )iµ . Then defining
F0 := (1 + p2 + q 2 )−1 lets us estimate
I∗ ≤ −hF0 , (v3,0,0 )2 + (v2,1,1 )2 iµ
1
+ h(v2,0,0 )2 , µ−1 ∂y2 (µF1 ) + ∂θ2 (v −2 F2 ) + µ−1 ∂y ∂θ (v −1 µF3 ) − ∂θ (v −2 F4 )iµ
2
≤ −(1 − ε)(Ω3,0 + Ω2,1 ) + ε1 Ω2,0 .
The first inequality results from applying Cauchy–Schwarz pointwise to the coefficients of the elliptic operator Av after integrating by parts. To get the second
inequality, we use estimates (4.4) and (4.5) from Theorem 1 to see that we can
bound F0 ≥ (1 − ε). We use Lemma 4 to control the remaining terms that came
from integrating by parts, noting that by taking κ0 and M sufficiently large, we
can by Condition [Ca] ensure that ε1 ∈ 0, a2 − ε . The result follows.
Arguments for the remaining Ωm,0 : The proofs here are essentially identical
to the corresponding cases considered in Lemmas 6.9, 6.11, and 6.12 of [13]. It is
important to note that we can use the results of Appendix D from [13] without
modification: because we did not need sharp estimates for derivatives of orders
three and higher there (as is also true here) we did not need to exploit the discrete
symmetry assumptions of that paper to get better results when integrating by
parts in θ. As a consequence, the conclusions of Lemmas D.1–D.7, which allow us
to estimate the nonlinear terms in the Ωm,n evolution equations, hold here without
modification. For the linear terms, the key estimate, which holds by Conditions [Ca]
and [C0i] and Theorem 1 when 3 ≤ m ≤ 5, is
1
I2 := h(v −2 − (m − 1)a)(vm,0,0 ), (vm,0,0 )iµ ≤ − Ωm,0 .
4
We omit further details.
Lemma 6. There exist 0 < ε < C < ∞ such that for (m, n) = (2, 1), one has
2
d
Ω2,1 ≤ − ε (Ω2,1 + Ω3,1 + Ω2,2 ) + Cβ 5 (Ω2,1 + Ω1,2 + Ω0,3 )
dτ
n
o
1
1
1
1
1
1
2
2
2
2
2
2
+ Cβ 2 Ω2,1
+ βΩ4,0
+ Ω3,1
+ Ω2,2
+ Ω1,3
+ Ω0,4
;
and for (m, n) ∈ {(3, 1), (2, 2)}, one has


X
1
d
Ωi,j 
Ωm,n ≤ − ε (Ωm,n + Ωm+1,n + Ωm,n+1 ) + Cβ 2 
dτ
4≤i+j≤5


X
1
2 
+ Cβ 2 
Ωi,j
;
4≤i+j≤5
and for (m, n) ∈ {(4, 1), (3, 2), (2, 3)}, one has


X
1
d
Ωm,n ≤ − ε (Ωm,n + Ωm+1,n + Ωm,n+1 ) + Cβ 2 
Ωi,j 
dτ
5≤i+j≤6


X
1
21
2 
+ Cβ 10 
Ωi,j
.
5≤i+j≤6
18
ZHOU GANG AND DAN KNOPF
Proof. We provide a detailed argument for Ω2,1 , and outline the remaining cases.
Argument for Ω2,1 : One has
1 d
Ω2,1 = I1 + I2 + I3 ,
2 dτ
with terms Ij that we now define. The first term on the rhs is
a
I1 := h(v2,1,1 ), Av (v2,1,1 )iµ ≤ Ω2,1 + I∗ ,
2
where
I∗ := h(v2,1,1 ), F1 ∂y2 (v2,1,1 )+F2 v −2 ∂θ2 (v2,1,1 )+F3 v −1 ∂y ∂θ (v2,1,1 )+F4 v −2 ∂θ (v2,1,1 )iµ .
The second term in
1 d
2 dτ Ω2,1
is
I2 := h(2v −2 − 2a)(v2,1,1 ), (v2,1,1 )iµ .
Below, we obtain a good estimate for I1 + I2 . Lemma D.1 of [13], which does
not require any symmetry assumptions, implies that the remaining contribution to
1 d
2 dτ Ω2,1 , which is
5
X
I3 :=
h(v2,1,1 ), E2,1,1,` iµ ,
`=0
can be estimated by
n 1
o
1
1
1
1
1
2
2
2
2
2
2
2
|I3 | ≤ Cβ 2 Ω2,1
+ βΩ4,0
+ Ω3,1
+ Ω2,2
+ Ω1,3
+ Ω0,4
+ Cβ 5 (Ω2,1 + Ω1,2 + Ω0,3 ) .
We now study I∗ . Define F0 := (1 + p2 + q 2 )−1 , as in the proof of Lemma 5.
Then by estimates (4.4) and (4.5) from Theorem 1, there exists ε1 > 0 such that
F0 ≥ (1 − ε1 ). Let γ ∈ (0, 1) be a parameter to be chosen. By integrating I∗
by parts, applying Cauchy–Schwarz pointwise to the coefficients of the quasilinear
elliptic operator Av , and using Lemma 4 to bound terms containing derivatives of
the coefficients of that operator, we find that there exists ε2 > 0 such that
I∗ ≤ −hF0 , {∂y (v2,1,1 )}2 + {v −1 ∂θ (v2,1,1 )}2 iµ + ε2 Ω2,1
n
o
≤ −(1 − ε1 ) γ k∂y (v2,1,1 )k2µ + kv −1 ∂θ (v2,1,1 )k2µ + (1 − γ)kv −1 ∂θ (v2,1,1 )k2µ
+ ε2 Ω2,1 .
Integrating by parts in y and using Conditions [C1] and [C2] shows that
3
k∂y (v2,1,1 )k2µ ≥ Ω3,1 − Cβ 5 Ω2,1 .
Integrating by parts in θ again using Conditions [C1] and [C2] shows that
3
kv −1 ∂θ (v2,1,1 )k2µ ≥ Ω2,2 − Cβ 2 Ω2,1 .
Then using Condition [Cr] along with the fact that ∂y2 ∂θ v is orthogonal to constants
in L2 (S1 ), we integrate by parts in θ to get
3
kv −1 ∂θ (v2,1,1 )k2S1 ≥ kv −2 (∂y2 ∂θ2 v)k2S1 − Cβ 2 k(v2,1,1 )k2S1
4
3
1−δ
≥
kv −2 (∂y2 ∂θ v)k2S1 − Cβ 2 k(v2,1,1 )k2S1 .
1+δ
Combining these estimates, we obtain ε3 , ε4 such that
n
o
I∗ ≤ −(1 − ε3 ) γ Ω3,1 + Ω2,2 + (1 − γ)hv −2 (v2,1,1 ), (v2,1,1 )iµ + ε4 Ω2,1 .
UNIVERSALITY IN MCF NECKPINCHES
19
Hence taking γ small and using Conditions [Ca] and [C0i] with Assumption [A2],
we obtain ε such that
3 I1 + I2 ≤ h (1 + ε)v −2 + ε − a (v2,1,1 ), (v2,1,1 )iµ − ε Ω3,1 + Ω2,2
2 ≤ −ε Ω2,1 + Ω3,1 + Ω2,2 .
The result follows.
Arguments for the remaining Ωm,n : As we note in the proof of Lemma 5,
the conclusions of Lemmas D.1–D.7 from [13], which allow us to estimate the nonlinear terms in the Ωm,n evolution equations, hold here without modification. For
the linear terms, the key estimates are
3 m
1
I1 + I2 ≤ h
−
+ n v −2 −
+ ε vm,n,n , vm,n,n iµ − ε Ωm+n,n + Ωm,n+1
4
2
2
≤ −ε Ωm,n + Ωm+1,n + Ωm,n+1 .
Condition [Ca] implies the first estimate. The bound v −2 < 59 , which follows from
Condition [C0i] in the inner region and from estimate (4.2) in the outer region,
implies the second estimate above. This concludes the proof for all remaining cases
(m, n) ∈ {(3, 1), (2, 2), (4, 1), (3, 2), (2, 3)}.
Finally, we bound the evolutions of the Ωm,n with relatively “few” y-derivatives,
namely (m, n) ∈ {(1, 2), (0, 3), (1, 3), (0, 4), (1, 4), (0, 5)}. The arguments to prove
these estimates are those that are most changed from our prior work [13].
Lemma 7. There exist 0 < ε < C < ∞ such that for (m, n) ∈ {(1, 2), (0, 3)} one
has
3
d
Ωm,n ≤ − ε (Ωm,n + Ωm+1,n + Ωm,n+1 ) + Cβ 5 (Ω2,1 + Ω1,2 + Ω0,3 )
dτ
n
o
1
1
1
1
1
1
2
2
2
2
2
2
+ βΩ4,0
+ Cβ 2 Ωm,n
+ Ω3,1
+ Ω2,2
+ Ω1,3
+ Ω0,4
;
and for (m, n) ∈ {(1, 3), (0, 4)}, one has


1
d
Ωm,n ≤ − ε (Ωm,n + Ωm+1,n + Ωm,n+1 ) + Cβ 2 
Ωi,j 
dτ
4≤i+j≤5


X
1
2 
+ Cβ 2 
Ωi,j
;
X
4≤i+j≤5
and for (m, n) ∈ {(1, 4), (0, 5)}, one has


1
d
Ωm,n ≤ − ε (Ωm,n + Ωm+1,n + Ωm,n+1 ) + Cβ 2 
Ωi,j 
dτ
5≤i+j≤6


X
1
21
2 
+ Cβ 10 
Ωi,j
.
X
5≤i+j≤6
Proof. We begin with a detailed argument for Ω1,2 .
Argument for Ω1,2 : As in Lemma 6, we write
1 d
Ω1,2 = I1 + I2 + I3 .
2 dτ
20
ZHOU GANG AND DAN KNOPF
Here, the first term on the rhs is
I1 := h(v1,2,2 ), Av (v1,2,2 )iµ ≤
a
Ω1,2 + I∗ ,
2
where
I∗ := h(v1,2,2 ), F1 ∂y2 (v1,2,2 )+F2 v −2 ∂θ2 (v1,2,2 )+F3 v −1 ∂y ∂θ (v1,2,2 )+F4 v −2 ∂θ (v1,2,2 )iµ
is analyzed below. The second term in
1 d
2 dτ Ω1,2
is
I2 := h(3v −2 − 2a)(v1,2,2 ), (v1,2,2 )iµ .
We estimate I1 +I2 below. Lemma D.1 of [13], which does not require any symmetry
d
Ω1,2 , which is the
assumptions, implies that the remaining contribution to 12 dτ
P5
quantity I3 := `=0 h(v1,2,2 ), E1,2,2,` iµ , can be estimated by
n 1
o
1
1
1
1
1
3
2
2
2
2
2
2
|I3 | ≤ Cβ 2 Ω1,2
+ βΩ4,0
+ Ω3,1
+ Ω2,2
+ Ω1,3
+ Ω0,4
+ Cβ 5 (Ω2,1 + Ω1,2 + Ω0,3 ) .
To study I∗ , we again define F0 := (1 + p2 + q 2 )−1 . Then by estimates (4.4)
and (4.5) from Theorem 1, there exists ε1 > 0 such that F0 ≥ (1 − ε1 ). Let
γ ∈ (0, 1) be a parameter to be chosen. By integrating I∗ by parts, applying
Cauchy–Schwarz pointwise to the coefficients of the quasilinear elliptic operator
Av , and using Lemma 4 to bound terms containing derivatives of the coefficients of
that operator, we find that there exists ε2 > 0 such that
I∗ ≤ −hF0 , {∂y (v1,2,2 )}2 + {v −1 ∂θ (v1,2,2 )}2 iµ + ε2 Ω1,2
n
o
≤ −(1 − ε1 )γ k∂y (v1,2,2 )k2µ + kv −1 ∂θ (v1,2,2 )k2µ
Z
− (1 − ε1 )(1 − γ)
kv −1 ∂θ (v1,2,2 )k2S1 µ dy + ε2 Ω1,2 .
βy 2 ≤20
Conditions [C1] and [C2] let us integrate by parts in y to estimate k∂y (v1,2,2 )k2µ ≥
3
Ω2,2 − Cβ 5 Ω1,2 , and then integrate by parts in θ to estimate kv −1 ∂θ (v1,2,2 )k2µ ≥
3
Ω1,3 − Cβ 2 Ω1,2 . Thus we obtain ε3 such that
I∗ ≤ −(1 − ε1 )γ Ω2,2 + Ω1,3 + ε3 Ω1,2
Z
− (1 − ε1 )(1 − γ)
kv −3 ∂y ∂θ3 vk2S1 µ dy.
βy 2 ≤20
We now estimate I1 + I2 . Using Conditions [Ca], [C0i], and estimate (4.2), we
observe that I2 := h(3v −2 − 2a)(v1,2,2 ), (v1,2,2 )iµ may be estimated by 10
Z
19
2a
kv −3 ∂y ∂θ2 vk2S1 µ dy
I2 = − Ω1,2 +
3
10 βy2 ≤20
Z
Z 11 −2 4a 2
+
v −
(v1,2,2 ) dθ µ dy
10
3
βy 2 ≤20 S1
Z
Z 4a 2
+
3v −2 −
(v1,2,2 ) dθ µ dy
3
βy 2 >20 S1
Z
2a
19
≤ − Ω1,2 +
kv −3 ∂y ∂θ2 vk2S1 µ dy.
3
10 βy2 ≤20
10See below for the formula we use to get this decomposition for general (m, n).
UNIVERSALITY IN MCF NECKPINCHES
21
Combining this with the estimates above, we obtain
Z
a
Ω1,2 − (1 − ε1 )γ Ω2,2 + Ω1,3 +
I1 + I2 ≤ ε3 −
Ξ µ dy,
6
βy 2 ≤20
where the quantity that remains to be controlled is
19 −3
Ξ :=
kv ∂y ∂θ2 vk2S1 − (1 − ε1 )(1 − γ)kv −3 ∂y ∂θ3 vk2S1 .
10
To estimate Ξ, let P denote orthogonal projection onto span{e±iθ } ⊂ L2 (S1 ).
Then using Condition [Cr], it is easy to see that there exists ε4 depending only on
δ such that in the inner region,
kv −3 ∂y ∂θ3 vk2S1 ≥ (1 − ε4 ) kv −3 P(∂y ∂θ3 v)k2S1 + kv −3 (1 − P)(∂y ∂θ3 v)k2S1 .
Again using [Cr] together with the fact that (1 − P)(∂y ∂θ3 v) ⊥ span{1, e±iθ }, we
integrate by parts as in [13] to estimate that
kv −3 (1 − P)(∂y ∂θ3 v)k2S1 = kv −3 ∂θ {(1 − P)(∂y ∂θ2 v)}k2S1
≥ 4(1 − ε4 )kv −3 (1 − P)(∂y ∂θ2 v)k2S1 .
Then using the fact that ∂y ∂θ2 v ⊥ span{1} in L2 (S1 ), we obtain
o
n 19
Ξ≤
(1 + ε4 ) − (1 − ε4 )(1 − ε1 )(1 − γ) kv −3 P(∂y ∂θ2 v)k2S1
10
n 19
o
+
(1 + ε4 ) − 4(1 − ε4 )(1 − ε1 )(1 − γ) kv −3 (1 − P)(∂y ∂θ2 v)k2S1 .
10
It is easy to choose γ ∈ (0, 1) so that the second quantity in braces above is negative.
To control the remaining term, we observe that the new Condition [C2i] implies
that kP(∂y ∂θ2 v)kS1 . β 2 , and then use Cauchy-Schwarz to see that
Z
1
2
kv −3 P(∂y ∂θ2 v)k2S1 µ dy ≤ ρβ 2 Ω1,2
,
βy 2 ≤20
where ρ(M ) & 0 as M → ∞. The result follows.
Arguments for the remaining Ωm,n : The nonlinear terms are estimated as
above, mutatis mutandis. To control the linear terms, we first observe that
Z Z
I2 :=
(n + 1)v −2 − (m + n − 1)a (vm,n,n )2 dθ µ dy
R S1
Z
2a
26 + n − 9m
= − Ωm,n +
kv −(n+1) ∂ym ∂θn vk2S1 µ dy
3
10
βy 2 ≤20
Z
5
9(m + n) − 16 −2
v − m+n−
a (vm,n,n )2 dθ µ dy
+
10
3
βy 2 ≤20
Z
5
+
(n + 1)v −2 − m + n −
a (vm,n,n )2 dθ µ dy
3
2
βy >20
Z
2a
26 + n − 9m
≤ − Ωm,n +
kv −(n+1) ∂ym ∂θn vk2S1 µ dy.
3
10
2
βy ≤20
The estimate here again follows from Conditions [Ca], [C0i], and estimate (4.2).
Thus we obtain
Z
a
Ωm,n − (1 − ε1 )γ Ωm+1,n + Ωm,n+1 +
Ξ µ dy,
I1 + I2 ≤ ε3 −
6
βy 2 ≤20
22
ZHOU GANG AND DAN KNOPF
where the quantity that remains to be controlled is now
Ξ :=
26 + n − 9m −(n+1) m n 2
(n+1)
vk2S1 .
kv
∂y ∂θ vkS1 − (1 − ε1 )(1 − γ)kv −(n+1) ∂ym ∂θ
10
We separate Ξ by using orthogonal projection as above, using Condition [C2i] to
control the term involving P(∂ym ∂θn v), and integrating by parts in θ to control the
term involving (1 − P)(∂ym ∂θn v), using the fact that 26+n−9m
≤ 31
10
10 < 4 for all (m, n)
under consideration. This completes the proof.
5.2. Second-order estimates in the inner region. With Lemmas 5–7 in hand,
we are ready to prove Theorem 2.
We begin by deriving L2µ bounds on all derivatives of orders two through five.
Lemma 8. Suppose that a solution v = v(y, θ, τ ) of equation (2.3) satisfies Assumption [A1] at τ = 0, as well as [Ca], [C0]–[C3], [C0i]–[C2i], [Cg], [Cr], and
[Cs] for τ ∈ [0, τ1 ]. Then for the same time interval, one has
β 2 Ω2,0 + Ω1,1 + Ω0,2 + βΩ3,0 + Ω2,1 + Ω1,2 + Ω0,3 . β 4
and
4
β 5 Ω4,0 + Ω3,1 + Ω2,2 + Ω1,3 + Ω0,4 +
X
Ωm,n . β 4 .
m+n=5
Proof. The proof of Proposition 6.10 in [13] applies without change to establish the
first estimate.
To establish the second estimate, we define
X
4
Υ := β 5 Ω4,0 + Ω3,1 + Ω2,2 + Ω1,3 + Ω0,4 +
Ωm,n ,
m+n=5
noting the β-weight imposed on the first term. By Lemmas 5–7 and Cauchy–
Schwarz, there exist 0 < ε < C < ∞ such that
1
d
Υ ≤ −εΥ + C β 2 Υ 2 + β 4 .
dτ
Assumption [A6] bounds Υ at τ = 0. It follows that Υ . β 4 . The result follows. Using these estimates, we complete the proof of Theorem 2.
Proof of Theorem 2. Lemma 8 provides L2µ bounds on derivatives of orders two
through five. To complete the proof, we use Sobolev embedding to obtain pointwise
bounds on derivatives of orders two and three. Because of the weighted norm k · kµ
6
these pointwise bounds are not uniform in y. Using the facts that µ ∼ hyi− 5 as
1
|y| → ∞ and that |y| . β − 2 in the inner region, one obtains
1
3
β 2 (∂y2 v)2 + v −2 (∂y ∂θ v)2 + v −4 (∂θ2 v)2 . β 4− 10 − 5
and
1
3
β(∂y3 v)2 + v −2 (∂y2 ∂θ v)2 + v −4 (∂y ∂θ2 v)2 + v −6 (∂θ3 v)2 . β 4− 10 − 5
in the inner region. These inequalities are equivalent to estimates (4.7)–(4.8).
UNIVERSALITY IN MCF NECKPINCHES
23
6. The second bootstrap machine
In this section and those that follow, we analyze the asymptotic behavior of
solutions. Specifically, we show that a solution v may be decomposed into a slowlychanging “main component” and a rapidly-decaying “small component.” The main
component is controlled by finitely many parameters, some of which encode information about the optimal coordinate system at a given time.
We accomplish this by building a second bootstrap machine, following [13] and
making modifications where necessary. Many of the arguments of [13] apply here
and so do not have to be repeated in detail. Significant new arguments are needed
in two areas. (i) The discrete symmetry assumptions imposed in [13] effectively
fixed the cylindrical axis once and for all, which allowed us to track the behavior of
the main component of the solution using only two parameters. Here, we must deal
with seven parameters, corresponding to the seven-dimensional weakly unstable
eigenspace of the linearization of mcf, as explained in Section 2.3. (ii) We must
derive improved estimates for |v± | and |(∂y v)± | in the inner region, stated as (6.10)
below. These allow the first bootstrap machine to function without the discrete
symmetries imposed in our earlier work.
6.1. Inputs. The second bootstrap machine requires three sets of inputs.
First we require that u be a solution of equation (2.1) satisfying the following
condition:
[Cd] There exists t# > 0 such that for 0 ≤ t ≤ t# , there exist C 1 functions a(t),
b(t) and βk (t) (k = 0, . . . , 4) such that u admits a decomposition
(6.1)
u(x, θ, t) =λ(t)v(y, θ, τ )
"
1
2 + b(t)y 2 2
+ β0 (t)y + β1 (t) cos θ + β2 (t) sin θ
=λ(t)
2 − 2a(t)
#
+ β3 (t)y cos θ + β4 (t)y sin θ + φ(y, θ, τ )
with the L2 orthogonality properties
ay 2
e− 2 φ ⊥ 1, y, 1 − ay 2 , cos θ, sin θ, y cos θ y sin θ ,
where
a(t) := −λ(t)∂t λ(t),
y := λ−1 (t)x,
Z t
τ (t) :=
λ−2 (s) ds,
0
βk (t# ) = 0, (k = 0, . . . , 4),
1 1
a(t# ) = − b(t# ).
2 4
Condition [Cd] follows from our Main Assumptions by an implicit function theorem
argument highly analogous to the decomposition result proved in [14] and also used
in [13]. We omit the details here.
24
ZHOU GANG AND DAN KNOPF
To state the second set of inputs, we define majorizing functions to control the
decay of the quantities φ(y, θ, τ ), a(t(τ )), and b(t(τ )) that appear in equation (6.1).
These are:
(6.2)
Mm,n (T ) := max β −
m+n
1
2 − 10
τ ≤T
(τ )kφ(·, ·, τ )km,n ,
(6.3)
1 1
A(T ) := max β −2 (τ ) a(t(τ )) − + b(t(τ )) ,
τ ≤T
2 4
(6.4)
B(T ) := max β − 2 (τ )|b(t(τ )) − β(τ )|,
3
τ ≤T
where (m, n) ∈ {(3, 0), (11/10, 0), (2, 1), (1, 1)}. By standard regularity theory
for quasilinear parabolic equations, if the initial data satisfy the Main Assumptions
[A1]–[A7] for b0 and c0 sufficiently small, then, making t# > 0 smaller if necessary,
the solution will satisfy the second set of inputs for this bootstrap argument, which
are contained in the following condition:
[Cb] For any τ ≤ τ (t# ), one has
1
A(τ ) + B(τ ) + |M (τ )| . β − 20 (τ ),
|βk (t)| . β 2 (τ ), k = 0, 1, 2, 3, 4,
where M denotes the vector
M := (Mi,j ), (i, j) ∈ {(3, 0), (11/10, 0), (2, 1), (1, 1)} .
1
Remark 3. Condition [Cb] implies estimates on v and v − 2 ∂y v in the inner region
1
1
βy 2 ≤ 20, which in turn imply Condition [C0i], the estimate |∂y v| . β 2 v 2 of
1
Condition [C1i], and the estimate hyi−1 |∂y v| ≤ M 4 β of Condition [Cg], for M
sufficiently large.
The final inputs to this bootstrap machine are the outputs of the first bootstrap
machine, which are listed in Section 4.2. They follow from Theorems 1–4. In the
sequel, we give ourselves the freedom to use all estimates detailed in Section 4.2.
For the reader’s convenience, we list here the implications of those estimates most
commonly used in this machine: for any τ ∈ [0, τ (t# )], the quantity v satisfies
v(y, θ, τ ) ≥ 1;
(6.5)
and there exist constants 0 1 and C, independent of τ (t# ), such that
|∂y v| ≤ C,
(6.6)
|v −1 ∂θ2 v|
(6.7)
(6.8)
(6.9)
1
v −1 |∂y v| ≤ Cβ 2 ,
13
|∂y2 v| ≤ Cβ 20 ,
+ |∂y ∂θ2 v| ≤ 0 1,
23
|∂y3 v| ≤ Cβ 20 ,
33
v −2 |∂y ∂θ2 v| + v −1 |∂y ∂θ v| + v −1 |∂y2 ∂θ v| + v −2 |∂θ2 v| + v −3 |∂θ3 v| ≤ Cβ 20 .
6.2. Outputs. We now state the main outputs of the second bootstrap machine.
They serve two main purposes. (i) The estimates stated below improve some of
those in the inputs. Together with the local well-posedness of the solution, this
fact enables us to continue making bootstrap arguments in larger time intervals.
(ii) Most importantly, the estimates below provide almost sharp control on various
components of the solution, and hence a clear understanding of the geometry and
asymptotic behavior of the evolving surface.
UNIVERSALITY IN MCF NECKPINCHES
25
Theorem 5. Suppose that Conditions [Cd] and [Cb] and estimates (6.5)–(6.9) hold
in the interval τ ∈ [0, τ (t# )]. Then there exists a constant C independent of τ (t# )
such that in the same time interval:
(i) The parameters a, b, and βk , (k = 0, . . . , 4) satisfy the estimates
A(τ ) + B(τ ) ≤ C,
|β0 (t)| ≤ Cβ
|β1 (t)| + |β2 (t)| ≤ Cβ
|β3 (t)| + |β4 (t)| ≤ Cβ
13
5
18
5
13
5
,
,
.
(ii) The function φ satisfies improved estimates implied by the bound
|M (τ )| ≤ C.
2
(iii) In the region βy ≤ 20, one can estimate that
(6.10)
21
|v± | + |∂y v± | ≤ Cβ 10 .
The theorem will be reformulated in Section 7 below. In that section, we provide
a heuristic outline of the ideas used in its proof, with an emphasis on those that
differ from what we used in [13]. The technical details of the proof appear in the
appendices.
For the reader’s convenience, we reformulate some implications of the estimates
proved in Theorem 5 and Lemma 9 (below) in forms that are most useful for our
subsequent applications.
Corollary 1. Suppose that the assumptions of Section 6.1 hold in a time interval
0 ≤ τ ≤ τ (t# ). Then for that same time interval, the parameters a and b satisfy
1 b(t)
−
+ O b2 (t) ,
2
4
1
b(t) = 1 + o(1) −1
,
b0 + τ (t)
a(t) =
while the “small component” φ of the decomposition (6.1) satisfies
8
khyi−3 φkL∞ + khyi−2 ∂y φkL∞ + kv −2 ∂θ2 vkL∞ . b 5 .
Another consequence of the estimates in Theorem 5 and Lemma 9 gives control
on the sequence of optimal coordinate systems we construct.
Corollary 2. Let the assumptions of Section 6.1 hold for times 0 ≤ t ≤ t# ,
and suppose that (x0 , x1 , x2 )n and (x0 , x1 , x2 )n+1 are optimal coordinates at times
tn < tn+1 respectively, where [tn , tn+1 ] ⊆ [0, t# ]. Then there exist Φ ∈ R3 and
Ψ ∈ O(3) such that (x0 , x1 , x2 )n+1 = Φ + (x0 , x1 , x2 )n Ψ, where
|Φ| + |Ψ − I| . b2opt (tn ).
Furthermore, for t ∈ [tn , tn+1 ], the parameters b and λ satisfy the estimates
b(t) = bopt (tn ) 1 + O(b2opt (tn )) ,
λ(t) = λopt (tn ) 1 + O(b2opt (tn )) .
26
ZHOU GANG AND DAN KNOPF
7. Improved estimates for the decomposition
We begin by putting the equation into a better form. Instead of studying the
a 2
evolution equation for φ, we fix a gauge by means of a weight factor e− 4 y chosen
so that the linearization, to be obtained below, becomes a self-adjoint operator.
By Condition [Cd] in Section 6, there exists a (t-scale) time 0 < t# ≤ ∞ such
that the gauge-fixed quantity
a
w(y, θ, τ ) := v(y, θ, τ )e− 4 y
2
can be decomposed as
a
2
(7.1) w = wab (y)+e− 4 y [β0 y +β1 cos θ +β2 sin θ +β3 y cos θ +β4 y sin θ]+ξ(y, θ, τ ).
a
2
Here wab := Va,b e− 4 y , and
a 2
e− 4 y ξ ⊥ 1, y, y 2 , cos θ, sin θ, y cos θ, y sin θ ,
(7.2)
where the orthogonality is with respect to the L2 (S1 × R) inner product,11 and
(7.3)
βk (t# ) =0, (k = 0, 1, 2, 3, 4),
1 1
a(t# ) = − b(t# ).
2 4
The parameters a, b and βk , k = 0, 1, · · · , 4, are C 1 functions of t.12
We now derive the pde satisfied by the “small component” ξ(y, θ, τ ). Below, we
use this to derive the ode satisfied by the parameters a(τ ), b(τ ), and βk (τ ). Using
equations (7.1) and (2.3), one finds that ξ evolves by
(7.4)
∂τ ξ = −L(a, b)ξ + F (a, b, β) + N1 (a, b, ξ) + N2 (a, b, ξ) + N3 (a, b, ξ),
where L(a, b) is the linear operator
L(a, b) := −∂y2 +
2 − 2a
1
a2 + ∂τ a 2 3a
y −
−
− ∂θ2 .
2
4
2
2 + by
2
The remaining terms on the rhs of the evolution equation (7.4) are as follows.
The quantity F has two parts, F (a, b, β) := F1 (a, b) + F2 (a, b, β), which are
"
r
ay 2
1
2b
aby 2
2 + by 2
− 4
√
p
F1 (a, b) := e
+a
3 − √
2 − 2a
2 − 2a (2 + by 2 ) 2
2 − 2a 2 + by 2
#
p
r
1
2
2 − 2a
2 + by 2
2 bτ y
p
−
−
.
−
√
3 aτ
2
2
2 + by
(2 − 2a) 2
2 − 2a 2 + by
and
F2 (a, b, β) := e−
ay 2
4
h
−2 2
−2
∂y2 + Va,b
∂θ − ay∂y + a + Va,b
− ∂τ
i
[β0 y + β1 cos θ + β2 sin θ + β3 y cos θ + β4 y sin θ] .
11In the remainder of this paper, the inner product we use is h·, ·i. As defined in Section 2.1,
this is the standard (unweighted) inner product for L2 ≡ L2 (S1 × R).
12To simplify notation, we mildly abuse notation by writing a(τ ), b(τ ), and β (τ ) for a(t(τ )),
k
b(t(τ )), and βk (t(τ )), respectively. This will not cause confusion, as the original functions a(t),
b(t), and βk (t) are not needed in what follows.
UNIVERSALITY IN MCF NECKPINCHES
27
The first nonlinear term N1 is
N1 (a, b, ξ) := −
(7.5)
1 2 − 2a ay2 ˜2
e 4 ξ ,
v 2 + by 2
where
2
2
ay
ay
ξ˜ := e− 4 [e 4 ξ + β0 y + β1 cos θ + β2 sin θ + β3 cos θy + β4 sin θy].
The second nonlinear term N2 is
N2 (a, b, ξ) := − e−
ay 2
4
p2
∂ 2 v.
1 + p2 + q 2 y
The final nonlinear term N3 is
(7.6)
N3 (a, b, ξ) := N3,1 (a, b, ξ) + N3,2 (a, b, ξ),
where
N3,1 :=
−2
Va,b
1 2
−
∂ ξ,
2 θ
and
N3,2
:= v −2
2
1 + p2
−2
2
− ay4
−
V
∂
v
e
θ
a,b
1 + q 2 + p2
ay 2
ay 2
2pq
q
− e− 4 v −1
∂θ ∂y v + e− 4 v −2
∂θ v.
2
2
1+p +q
1 + p2 + q 2
7.1. The finite-dimensional part of the decomposition. We next derive and
estimate ode for the parameters a, b and βk , (k = 0, · · · , 4) by using the evolution
equation (7.4) for ξ and the orthogonality conditions (7.2). Before providing the
details of the technical proof, we illustrate the key ideas involved by heuristically
outlining how one derives and estimates an ode for β0 .
a 2
Taking the L2 (S1 × R) inner product of (7.4) with the function ye− 4 y yields
E
D
E
D
E
D
a 2
a 2
a 2
ye− 4 y , ∂τ ξ = − ye− 4 y , L(a, b)ξ + ye− 4 y , F (a, b, β) + · · · .
a
2
The fact that ξ ⊥ ye− 4 y in (7.2) implies that
D
E D
E a D
E
a 2
a 2
a 2
a 2
τ
y 3 e− 4 y , ξ . |aτ | khyi−3 e 4 y ξkL∞ .
ye− 4 y , ∂τ ξ = ∂τ ye− 4 y , ξ +
4
a
2
Noting that ye− 4 y is an eigenfunction of the self-adjoint operator
a2
1
1
−∂y2 − ∂θ2 + y 2 − 2a −
2
4
2
that constitutes the “main part” of the linear operator L(a, b) appearing in (7.4),
we obtain
D
E
a 2
a 2
ye− 4 y , L(a, b)ξ . |aτ | + b khyi−3 e 4 y ξkL∞ .
a
2
Because ye− 4 y is an odd function of y and is independent of θ, we then get
Z ∞
D
E
a 2
a 2
ye− 4 y , F (a, b, β) = 2π [Ω1 (a, b)β0 − ∂τ β0 ]
y 2 e− 2 y dy,
−∞
where Ω1 is the positive scalar function
R ∞ 2 −2 − a y2
y V e 2 dy
1
R ∞ a,b− a y2
(7.7)
Ω1 (a, b) := −∞
≈ .
2
2
y e 2 dy
−∞
28
ZHOU GANG AND DAN KNOPF
q
2
1
1
Here we used the fact that Va,b = 2+by
2−2a ≈ 2 , which is implied by a ≈ 2 and the
fact that b is positive and small. Collecting the estimates above yields the desired
equation for β0 , which has the form
a 2
∂τ β0 − Ω1 (a, b)β0 = O (|aτ | + |b|)khyi−3 e 4 y ξkL∞ + · · · .
Finally, we sketch how one derives an estimate for β0 based on this ode. Using
the boundary condition β0 (τ (t# )) = 0 from (7.3), we can rewrite the equation for
β0 at any τ ≤ τ (t# ) as
Z τ (t# ) n R τ (t )
o
#
a 2
Ω1 (a,b)(s1 ) ds1
O (|aτ | + |b|)khyi−3 e 4 y ξkL∞ (s) + · · · ds.
β(τ ) =
e− s
τ
a
2
Using the fact that Ω1 ≈ 21 and the smallness of |aτ |, b, and khyi−3 e 4 y ξkL∞ , this
implies the desired estimate.
Using the method outlined above, we obtain the following estimates.
Lemma 9. For all times that the assumptions of Section 6.1 hold, one has
(7.8)
(7.9)
−
5
2
∂τ a + 2b + 4(2a − 1) = O(β 2 ),
1−a
5
∂τ b + b2 = O(β 2 ),
5
(7.10)
∂τ β0 = Ω1 (a, b)β0 + O(β 2 ),
(7.11)
∂τ β1 = aβ1 + (Small)1 ,
∂τ β2 = aβ2 + (Small)2 ,
∂τ β3 = (Small)3 ,
∂τ β4 = (Small)4 ,
where the terms (Small)` , (` = 1, . . . , 4), satisfy
n 16 18
o
ay 2 31 |(Small)` | . min β 5 , β 5 + β 20 hyi−5 ∂θ ξe 4 L∞ .
Furthermore, one has
1 (7.12)
2a − 1 + b . β 2
2
and
3
|b − β| . β 2 ,
(i.e. A + B . 1) and
(7.13)
(7.14)
5
|β0 | . β 2 ,
n 16
o
18
|β1 | + |β2 | . min β 5 , (1 + M4 )β 5 ,
n 11
o
13
|β3 | + |β4 | . min β 5 , (1 + M4 )β 5 .
The function M4 above is defined as
ay 2 21 M4 (τ ) := max β − 10 hyi−5 ∂θ ξ(τ )e 4 0≤s≤τ
We prove the lemma in Appendix A.
L∞
.
UNIVERSALITY IN MCF NECKPINCHES
29
7.2. The infinite-dimensional part of the decomposition. We next show how
to control the infinite-dimensional part ξ. Many of the arguments here are virtually
identical to their counterparts in [13]. Before providing details, we again begin with
a heuristic outline of the main ideas behind the proof.
We apply Duhamel’s principle to equation (7.4) for ξ to obtain
Z τ
U (τ, σ) · · · dσ,
ξ(τ ) = U (τ, 0)ξ(0) +
0
where U (τ, σ) is the propagator generated by the linear operator −L(a, b) from time
σ to τ . To prove the desired decay estimates for ξ, we exploit the decay of U (τ, σ)
in a suitable subspace of L∞ . To achieve this, we must overcome two difficulties.
(i) The generator of U (τ, σ) namely L(a, b), has a seven-dimensional nonpositive
a 2
a 2
a 2
a 2
a 2
eigenspace, spanned by e− 4 y , ye− 4 y , (ay 2 − 1)e− 4 y , (cos θ)e− 4 y , (sin θ)e− 4 y ,
a 2
a 2
(y cos θ)e− 4 y , and (y sin θ)e− 4 y . This implies that U (τ, σ) may grow in these
directions. (ii) The operator L(a, b) is not autonomous, so U (τ, σ) 6= e−tL(a,b) .
We overcome the first difficulty by using the fact that ξ is orthogonal to the
unstable eigenspace. At least intuitively, one expects U (τ, σ) to decay exponentially
fast in directions orthogonal to that space. To overcome the second difficulty,
we make another gauge change, reparameterizing the function ξ to obtain a new
function whose evolution is dominated by an autonomous operator.
Using these ideas, one first proves the following.
Lemma 10. For all times that the assumptions of Section 6.1 hold, one has
|Mm,n | . 1, for (m, n) ∈ (3, 0), (11/10, 0), (2, 1), (1, 1) .
The proof is essentially identical to that of Proposition 9.2 from [13], because it
relies only on the outputs of the first bootstrap machine, hence does not need the
discrete symmetry assumptions that were in force there. So we omit it here.
Next we turn to estimating the functions v± and (∂y v)± defined by formula (2.9).
These estimates represent a departure from [13]. In what follows, we present the
difficulties and the ideas used to overcome them when estimating v± . The arguments
for (∂y v)± are very similar, hence omitted. Our objective is to prove the following.
Lemma 11. For all times that the assumptions of Section 6.1 hold, M4 . 1, i.e.
ay 2 21
−5
(7.15)
hyi ∂θ ξe 4 ∞ . β 10 .
L
Lemma 12. For all times that the assumptions of Section 6.1 hold, in the region
βy 2 ≤ 20, one has
ay2 ay 2
21
4
(7.16)
e ξ± + (∂y e 4 ξ)± . β 10 .
Lemmas 11 and 12 are proved in Appendices B and C, respectively.
The difficulties encountered in their proofs are as follows. The decomposition of
v implies that
ay 2
|v± | ≤ |β1 | + |β2 | + |β3 y| + |β4 y| + |ξ± |e 4 ,
where the θ-independent functions ξ± are defined using (2.9). For the purposes of
11
Lemmas 11 and 12, the estimates |β3 | + |β4 | = O(β 5 ) from (7.14) are obviously
8
not good enough. In the region by 2 ≤ 20, they only give |βk y| . β 5 , which is
30
ZHOU GANG AND DAN KNOPF
21
significantly slower than the desired β 10 decay. The estimates for ξ in Lemma 10
are not good enough either. As noted above, the reason we want to prove stronger
decay is so that the first bootstrap machine can function without needing discrete
symmetry hypotheses. So we must derive improved estimates for |∂θ ξ|. But it is not
difficult to see that ∂θ ξ admits better decay estimates than does ξ. Differentiation
allows us to remove the slowly-decaying θ-independent components of the solution,
and thereby to obtain improved estimates.
In the remainder of this section, we outline the main ideas used in proving
estimate (7.16) in Appendix C. The steps used to prove (7.15) in Appendix B are
similar, hence will not be discussed here.
Using (7.4) and recalling definition (2.9) , we compute that
∂τ ξ± = −L0 (a)ξ± + · · · ,
where
a2 + ∂τ a 2 3a
y − .
4
2
Our strategy for estimating ξ± is similar to that used to estimate ξ in the proof
of Lemma 10, and hence has similar difficulties: the linear operator L0 (a) is timeL0 (a) := −∂y2 +
ay 2
ay 2
dependent and has nonpositive eigenvalues with eigenvectors e− 4 and ye− 4 .
To take care of these eigendirections, we use the orthogonality conditions imposed
2
2
on ξ in (7.2) to see that ξ± ⊥ span{e−ay /4 , ye−ay /4 }. The intuition behind
the argument is that we obtain good estimates for ξ± by applying L0 (a) to the
orthogonal complement of the finite-dimensional unstable subspace. We make this
rigorous in Appendix C. Once that work is done, our final result follows readily:
Proof of estimate (6.10) in Theorem 5. We use the decomposition of v in (6.1) to
relate ξ± to v± , obtaining
|v± | ≤ |β1 | + |β2 | + |β3 y| + |β4 y| + |ξ± |e
ay 2
4
.
The estimate for ξ± in Lemma 12 and the estimates for βk (k = 1, . . . , 4) in Lemma 9
21
then imply that |v± (y, θ, τ )| . β 10 (τ ).
The estimate for (∂y v)± is obtained similarly.
8. Proof of the Main Theorem
In this section, we collect the remaining arguments needed to complete the proof
of our Main Theorem, modulo the technical details collected in the appendices.
Proof of the Main Theorem. By placing an Angenent self-similarly shrinking torus
around the (approximate) center of the neck, one sees easily that the solution must
become singular before some time T ∗ < ∞. So suppose that [0, T∗ ) is the maximal
time interval such that for any time t ∈ [0, T∗ ) ⊆ [0, T ∗ ), we can construct an
optimal coordinate system in which
)
(s
2 + bopt (t)y 2
(8.1)
u(x, θ, t) = λopt (t)
+ φopt (y, θ, t) ,
1 + 12 bopt (t)
where y = λ−1
opt x, and φ satisfies the orthogonality conditions of Definition 1, along
with the estimate
−3
hyi φopt (·, ·, t) ∞ . b8/5
opt (t).
L
UNIVERSALITY IN MCF NECKPINCHES
31
In this case, we claim that for any sequence of times tn % T∗ at which we construct
optimal coordinate systems, one has bopt (tn ) → 0 and λopt (tn ) → 0 as n → ∞.
By the estimates in Theorem 5 (see also Corollaries 1–2) for the components of the
decomposition (8.1), this implies that the surface must become singular as t % T∗ .
We prove the claim by contradiction, showing that if either quantity bopt (tn ) or
λopt (tn ) has a positive lower bound, then the other does also, which implies that the
solution can be extended past T∗ , contradicting the assumption that T∗ is maximal.
We provide a detailed argument for the case that there exists a constant c∞ > 0
such that bopt (tn ) ≥ c∞ . (An analogous argument works if λopt (tn ) ≥ c0∞ .)
With respect to an optimal coordinate system constructed at tn , there exists a
time interval [tn−1 , tn ] ⊆ [0, tn ] in which the solution v = λ−1 u can be parameterized as in (2.10). Using the fact that ∂τ log λ = −a, and the boundary conditions
stipulated in Definition 1 that ensure that λopt = λ at any times at which we
construct optimal coordinates, we see that
λopt (tn ) = e
−
R tn
tn−1
a(τ̂ ) dτ̂
λopt (tn−1 ).
Then using the upper bound for b in Corollary 1, the consequence of estimate (7.12)
that a = 21 + O(b) is bounded independently of n, and the upper bound T∗ ≤ T ∗ ,
we conclude that there exists c > 0 independent of n such that λopt (tn ) ≥ cλopt (0).
Because n was arbitrary, this contradicts the maximality of T∗ and proves the claim.
Part (i) of the theorem follows directly from the claim.
Part (ii) then follows from Corollary 2, because the claim implies that we can
construct optimal coordinate systems up to the singular time.
Proving Part (iii) takes more work. Obtaining asymptotics for the sequential
parameters λopt (tn ) and bopt (tn ) that determine the “main components” of the
solution’s decomposition is complicated by the fact that we do not have ode for
them; we only have ode for the quantities λ and b that depend smoothly on the
sequential choices of λopt (tn ) and bopt (tn ). So we proceed as follows. Working in a
time interval [tn , tm ], we use the relation λ∂t λ = −a to see that
Z tm
2
2
λ (t) = λopt (tm ) +
2a(s) ds.
t
By the estimates for a in Corollary 1 and for b in Corollary 2, this implies that
λ2opt (tn ) = 1 + O(bopt (tm ) λ2opt (tm ) + (tm − tn ) .
Letting m → ∞ and using the fact proved above that λopt (tm ) & 0 as m → ∞, we
conclude that the asymptotic behavior of λopt (tn ) as n → ∞ is
p
(8.2)
λopt (tn ) = 1 + o(1) T − tn .
dτ
Next, by the estimate for a in Corollary 1 and the fact that dλ
= − a1 , we find that
as n → ∞, one has
1
τ (tn ) = 1 + o(1) log
.
T − tn
Then using the estimates for b and bopt in Corollaries 1–2, it follows that as n → ∞,
−1
1
.
(8.3)
bopt (tn ) = 1 + o(1) log
T − tn
Equations (8.2) and (8.3) establish Part (iii) of the theorem.
32
ZHOU GANG AND DAN KNOPF
Part (iv) of the theorem follows directly from the outputs of the first and second
bootstrap machines, as stated in Sections 4.2 and 6.2, respectively.
The proof is complete.
Appendix A. Proof of Lemma 9
In this appendix, we prove Lemma 9. To avoid unenlightening repetition, we
provide detailed arguments only for the a, b, β0 , and β3 evolution equations and
their estimates. The arguments for the others are almost identical.
A.1. Proofs of estimates (7.8) and (7.9). The derivations of (7.8) and (7.9) are
almost identical to those in our previous work [13], hence are only sketched here.
We rewrite F1 (a, b) in the form
ay 2
1
2b
1
1
aτ by 2
2
p
+
4a
−
2
−
b
y
−
a
−
F1 (a, b) = e− 4 √
τ
τ
2
1−a
2 − 2a
2 − 2a 2 + by 2 2 + by 2
2
ay
1
1
p
= e− 4 √
Γ1 − Γ2 y 2 + Γ 3 ,
2
2
2 − 2a 2 + by
with Γk (k = 1, 2, 3) defined as
Γ1 := b + 4a − 2 −
1
aτ ,
1−a
Γ2 := b2 + ∂τ b,
Γ3 :=
aτ by 2
b3 y 4
−
.
2(2 + by 2 ) 2 − 2a
a
2
a
2
Then we take an inner product of (7.4) with the functions e− 4 y and (ay 2 −1)e− 4 y ,
applying the orthogonality conditions of (7.2) to obtain
5
|Γ1 | + |Γ2 | . β 2
(A.1)
Recalling the definitions of Γ1 and Γ2 above, it is easy to see that this estimate
implies (7.8) and (7.9).
A.2. Proof of the estimates in (7.12).
Proof of the estimates in (7.12). We write Γ1 as
1
1
1
Γ1 =
2Γ̃1 − ∂τ Γ̃1 + ∂τ b − 2Γ̃21 + b2 ,
1−a
4
8
with Γ̃1 := a − 21 + 14 b. To estimate the various components, we use the assumptions
3
1
max |b(s) − β(s)|β − 2 (s) = B(τ ) . β − 20 (τ )
s≤τ
and
5
1 1 max a − + b β − 2 (s)
s≤τ
2 4
to obtain b ≤ 2β and |Γ̃1 | . β 2 . Hence we get
(A.2)
|∂τ Γ̃1 − 2Γ̃1 | . β 2 .
By the boundary condition a(τ (t# )) =
1
2
− 41 b(τ (t# )) in (7.3), we have
Γ̃1 (τ# ) = 0,
UNIVERSALITY IN MCF NECKPINCHES
33
where we denote τ (t# ) by τ# . We rewrite (A.2) as
Z τ#
|Γ̃1 (τ )| .
e2(τ −s) β 2 (s) ds
τ
Z τ#
(A.3)
. β 2 (τ )
e2(τ −s) ds
τ
. β 2 (τ ),
and then we use the definition of A in (6.3) to obtain the first estimate in (7.12).
The argument used to estimate b is identical to the corresponding argument
from [13]. We rewrite Γ2 and use estimate (A.1) to see that
h1
1
1
1 i
−
. β2.
∂τ − 1 = ∂τ
b
b β
−1
. It is clear that 1b − β1 τ =0 = 0, and consequently
Recall that β := b−1
0 +τ
that
Z τ
1
1
1
1 −
)
.
β 2 (τ ) dτ . β − 2 (τ ).
(τ
b β
0
3
Finally, recalling the definition of B in (6.4), we obtain |b(τ ) − β(τ )| . β 2 (τ ) and
B . 1.
The following facts will be used frequently in the rest of the paper.
Lemma 13. For all times that the assumptions in Section 6.1 hold, one has
|aτ | + |bτ | . β 2 ,
(A.4)
and
1
1−β4 ≤
(A.5)
1
b
≤ 1 + β4.
β
Proof. To obtain estimate (A.4), we combine the estimate for bτ from (A.1), the
estimate for aτ from (A.2), and the estimate Γ̃1 = O(β 2 ) from (A.3).
Estimate (A.5) follows from the observation that
3
max |b(s) − β(s)|β − 2 (s) = B(τ ) . 1.
s≤τ
A.3. Proofs of estimates (7.10) and (7.13). We start with proving (7.10), using
methods outlined in the discussion before Lemma 9.
Proof of estimate (7.10). Following the approach outlined in our introduction to
Lemma 9, we take the inner product of (7.4) with ye−
X
5
ay 2
(A.6)
ye− 4 , ∂τ ξ =
Ak ,
ay 2
4
to obtain
k=1
where the terms on the rhs are defined by
1
2
ay 2
+a
a2 + aτ 2 3a 1 2
− ay4
A1 := − ye− 4 , −∂y2 +
y −
− ∂θ − 2
ξ
+
ye
,
N
3,1
4
2
2
2 + by 2
1
ay 2
+a
a2 + aτ 2 3a
−2 2
= − ye− 4 , −∂y2 +
y −
− Va,b
∂θ − 2
ξ ,
4
2
2 + by 2
34
ZHOU GANG AND DAN KNOPF
and
D
E
ay 2
A2 := ye− 4 , F (a, b, β) ,
D
E
ay 2
A3 := ye− 4 , N1 ,
D
E
ay 2
A4 := ye− 4 , N2 ,
D
E
ay 2
A5 := ye− 4 , N3,2 .
We now show how the terms in equation (A.6) are controlled.
For the lhs of (A.6), we use the orthogonality condition ξ ⊥ ye−
obtain
E
D
E a D
E
D
ay 2
ay 2
ay 2
τ
y 3 e− 4 , ξ
ye− 4 , ∂τ ξ = ∂τ ye− 4 , ξ +
4
2
−3 ay4
= O |aτ |khyi e ξkL∞
ay 2
4
in (7.2) to
71
= O(β 20 ).
Then we use the estimate |aτ | = O(β 2 ) from (A.4), and the assumption that M3,0 .
1
β − 20 to obtain
8
31
−3 ay42 . β 5 M3,0 . β 20 .
hyi e ξ L∞
ay 2
ay 2
In A1 , many terms cancel because of the fact that ξ ⊥ ye− 4 , where ye− 4
2
−2 2
is an eigenfunction of the self-adjoint operator −∂y2 + a4 y 2 − Va,b
∂θ . We compute
that
1
D
ay 2
+a E
a2
3a
−2 2
A1 = − ye− 4 , [−∂y2 + y 2 −
− Va,b
∂θ − 2
]ξ
4
2
2
1
i E
D
ay 2
aτ 2 E D − ay2 h 12 + a
2 +a
y ξ + ye 4 ,
−
ξ
− ye− 4 ,
4
2 + by 2
2
D
D
E
ay 2
ay 2
aτ 2 E 1
y2
= − ye− 4 ,
y ξ −
+ a b ye− 4 ,
ξ .
2
4
2
2(2 + by )
Hence we conclude that
|A1 | . (|aτ | + b)khyi−3 e
ay 2
4
51
ξkL∞ . β 20 .
ay 2
The expression for A2 can be simplified by observing that ye− 4 is odd in y and
independent of θ, hence is orthogonal to the functions even in y or θ-dependent.
Thus we compute that
D
E
ay 2
A2 = ye− 4 , F1 (a, b) + F2 (a, b, β)
D
h
i
E
ay 2
ay 2
−2
= ye− 4 , e− 4 ∂y2 − ay∂y + a + Va,b
− ∂τ β 0 y .
Consequently, we get
A2 = 2π Ω1 (a, b)β0 − ∂τ β0
Z
∞
−∞
where Ω1 (a, b) is the constant defined in (7.7).
y 2 e−
ay 2
2
dy,
UNIVERSALITY IN MCF NECKPINCHES
35
For A3 , we use the definition of ξ˜ in (7.5) and the assumptions in Condition [Cb]
that |βk | . β 2 for k = 0, 1, . . . , 4 to obtain
ay 2
A3 . hyi−6 e 2 ξ˜2 ∞
L
4
2
X
ay 2
. hyi−3 e 2 ξ 2 ∞ +
|βk |2
(A.7)
L
k=0
3
.β .
For A4 , direct calculation yields
|A4 | . khyi−1 pk2L∞ k∂y2 vkL∞ = khyi−1 ∂y vk2L∞ k∂y2 vkL∞ .
The decomposition of v implies that
∂y v = √
n ay2 o
by
p
+ β0 + β3 cos θ + β4 sin θ + ∂y e 2 ξ .
2 − 2a 2 + by 2
Hence we get
h ay2 i
−1
−1
hyi ∂y v ∞ ≤ b + |β0 | + |β3 | + |β4 | + ∂
e 2 ξ (A.8)
hyi
y
L
L∞
Recall that khyi−1 ∂y [e
ay 2
4
have ke ξk1,1 . β
implies that
21
20
ay 2
2
ξ]kL∞ = ke
ay 2
2
. β.
ξk1,1 , and that by assumption on M1,1 we
13
. This, together with the estimate |∂y2 v| = O(β 20 ) in (6.8)
53
|A4 | . β 20 .
We use the definition of the quantity N3,2 introduced in (7.6) to decompose A5
into four terms,
h
i
D
E
ay 2
ay 2
−2
A5 = ye− 4 , v −2 − Va,b
∂θ2 ξe− 4
D
2 E
ay 2
q2
2 − ay4
− ye− 4 , v −2
∂
ξe
θ
1 + p2 + q 2
D
2 E
ay 2
2pq
− ay4
+ ye− 4 , v −1
∂
∂
ve
θ y
(A.9)
1 + p2 + q 2
D
E
2
ay
ay 2
q
+ ye− 4 , v −2
∂θ ve− 4
2
2
1+p +q
4
X
=
A5,` ,
`=1
where the various terms A5,` (` = 1, . . . , 4) are naturally defined.
For A5,1 , we integrate by parts in θ to remove the slowly decaying θ-independent
−2
components in v −2 − Va,b
. Thus we compute that
D
E
ay 2
A5,1 = 2 ye− 2 , v −3 ∂θ v∂θ ξ
and hence can estimate
ay 2
|A5,1 | . kv −2 ∂θ vkL∞ hyi−3 e 4 ∂θ ξ L∞
.
36
ZHOU GANG AND DAN KNOPF
We relate e
ay 2
4
∂θ ξ to ∂θ v using the decomposition of v in (6.1), obtaining
hyi−2 e
ay 2
4
|∂θ ξ| ≤ hyi−2 |∂θ v| +
4
X
|βk |
k=1
(A.10)
. v −2 |∂θ v| +
4
X
|βk |
k=1
33
. β 20 .
33
Here we used the estimate v −2 |∂θ v| . β 20 from (6.9) along with the fact that
(A.11)
hyi−1 . v −1 ,
equivalently,
v . hyi.
This, in turn, is implied by three facts: the computation that
Z y
|v(y, θ, τ )| ≤ |v(0, θ, τ )| +
|∂z v(z, θ, τ )| dz,
0
1
the consequence v(0, θ, τ ) . 1 of our input assumption that M3,0 . β − 20 , and the
assumption that |∂y v| . 1 from (6.6). We collect the estimates above to obtain
33
|A5,1 | . β 10 .
P4
Turning to `=2 A5,` , we observe that each of these terms contains a rapidly
decaying factor q = v −1 ∂θ v. We apply (A.11) again to get
33
khyi−1 qkL∞ . kv −2 ∂θ vkL∞ . β 20 .
This, together with the estimates in (6.9), implies that
4
X
16
A
5,` . β 5 .
`=2
Collecting the estimates above completes the proof of estimate (7.10).
Proof of estimate (7.13). Let τ# denote τ (t# ), and recall the boundary condition
β0 (τ# ) = 0 from (7.3). We use (7.10) to write
Z τ# R τ
#
13
|β0 (τ )| .
e− κ Ω(a,b)(s) ds β 5 (κ) dκ
τ
Then we use the consequence of (7.7) that Ω(a, b) ≥ 1/2 to conclude that
|β0 (τ )| . β
13
5
(τ ).
A.4. Proofs of estimates (7.11) and (7.14).
Proof of estimate (7.11). As we did in deriving (A.6), we take the inner product of
(7.4) with cos θe−
ay 2
4
to obtain
D
cos θe−
ay 2
4
5
E X
, ∂τ ξ =
Ãk ,
k=1
where the terms Ãk are defined like those in (A.6), replacing y by cos θ where
needed.
UNIVERSALITY IN MCF NECKPINCHES
37
The estimates for the various terms are very similar to those in the proof of
estimate (7.10). The only difference is that the presence of the factor cos θ here
allows us to integrate by parts in θ to remove some slowly-decaying θ-independent
parts. In what follows, we estimate Ã1 , Ã2 , and Ã4 in detail to illustrate the main
ideas. We omit detailed proofs of the estimates for the other terms.
For Ã1 , we have
h
D
i E
ay 2
a2 + aτ 2 3a
−2 2
−2
Ã1 = − cos θ e− 4 , − ∂y2 +
y −
− Va,b
∂θ − Va,b
ξ
4
2
D
E
ay 2
aτ
cos θ y 2 e− 4 , ξ
=−
4
E
ay 2
aτ D
sin θ y 2 e− 4 , ∂θ ξ .
=
4
−2 2
−2
In the second step, we used the simple observation [Va,b
∂θ + Va,b
] cos θ = 0; and in
the last step, we integrated by parts in θ. This calculation directly implies that
o
n
ay 2
ay 2 |Ã1 | . |aτ | min hyi−3 e 4 ξ ∞ , hyi−5 e 4 ∂θ ξ ∞
L
L
o
n 7
ay 2
2
−5
. min β 2 , β hyi e 4 ∂θ ξ ∞ .
L
For Ã2 , one can use symmetries to cancel many terms, as we did in Section A.3
in the proof of estimate (7.10). Here we get
D
h
i
E
ay 2
−2 2
−2
Ã2 = cos θ e− 2 , ∂y2 + Va,b
∂θ − ay∂y + a + Va,b
− ∂τ β1 cos θ
Z 2π Z ∞
ay 2
= [aβ1 − ∂τ β1 ]
cos2 θ e− 2 dy dθ.
0
−∞
For Ã4 , we integrate by parts in θ to remove the θ-independent components,
yielding
D
E
D
E
ay 2
ay 2
Ã4 := cos θe− 4 , N2 = − sin θe− 4 , ∂θ N2 .
Using the definition of N2 , this becomes
D
2
2
ay 2
ay 2
2p∂θ p
2
− ay4 2p [p∂θ p + q∂θ q] 2
Ã4 = − sin θe− 4 , e− 4
∂
v
−
e
∂ v
y
1 + p2 + q 2
(1 + p2 + q 2 )2 y
E
ay 2
p2
2
+ e− 4
∂
∂
v
θ
y
1 + p2 + q 2
= K1 + K2 + K3 ,
where the terms K` (` = 1, 2, 3) are naturally defined.
It is easy to estimate K2 and K3 . Indeed, using the estimates in (6.9) and the
estimate hyi−1 . v −1 from (A.11), we get
18
|K` | . b 5
for ` = 2, 3. Estimating K1 is more involved. We begin by integrating by parts in
the y variable, using 2p∂y2 v = ∂y (∂y v)2 to obtain
D h
E
ay 2
∂θ ∂y v i
2
K1 = ∂y sin θe− 2
,
(∂
v)
.
y
1 + p2 + q 2
38
ZHOU GANG AND DAN KNOPF
We bound the term hyi−2 (∂y v)2 by β 2 as in (A.8). Then we use (6.9) to see that
ay 2
8
∂ ∂ v the terms in ∂y sin θe− 2 1+pθ 2 y+q2 are of order β 5 . Thus we get
|K1 | . β
18
5
.
We collect the estimates obtained above to conclude that
|Ã4 | . β
18
5
.
For Ã3 and Ã5 , we apply similar methods to conclude that
o
n 16
ay 2
31 |Ã3 | + |Ã5 | . min β 5 , β 20 hyi−5 e 4 ∂θ ξ .
∞
L
This completes our proof of estimate (7.11).
Proof of estimate (7.14). By the boundary condition (7.3), we have β1 (τ# ) = 0,
where we again define τ# = τ (t# ). We rewrite estimate (7.11) in the form
n 16
o
18
|∂τ β1 − aβ1 | . min β 5 , (M4 + 1)β 5 .
This implies that
τ#
Z
|β1 (τ )| .
e−
R τ#
κ
a(s) ds
o
n 16
18
· min β 5 , (M4 + 1)β 5 (κ) dκ.
τ
Using the fact that a ≥ 1/4, we thus conclude that
n 16
o
18
|β1 (τ )| . min β 5 (τ ), (M4 (τ ) + 1)β 5 (τ ) .
Appendix B. Proof of Lemma 11
Proof of Lemma 11. By differentiating equation (7.4), we find that ∂θ ξ evolves by
3
X
∂τ (∂θ ξ) = −L0 (a) ∂θ ξ +
Dk ,
(B.1)
k=1
where the terms on the rhs are defined by
1
a2 + ∂τ a 2 a
y − − 1 − ∂θ2 ,
L0 (a) := −∂y2 +
4
2
2
2 − 2a
D1 := 1 − a −
∂θ ξ,
2 + by 2
D2 := ∂θ F (a, b, β),
D3 :=
3
X
∂θ N` .
`=1
Recall that the nonlinear terms N` appearing above are defined in Section 7.
Our first observations are the following key facts.
Lemma 14. For all times that the assumptions in Section 6.1 hold, one has the
orthogonality condition
(B.2)
a
2
e− 4 y ∂θ ξ ⊥ {1, y, y 2 , cos θ, sin θ, y cos θ, y sin θ},
UNIVERSALITY IN MCF NECKPINCHES
39
along with estimates
3
X
−5 a y2 hyi e 4 Dk ∞ . β 13
5 .
L
(B.3)
k=1
This lemma is proved in Section B.1 below.
In what follows, we pursue the strategies outlined in Section 7. Our next step is
to put (B.1) into a more convenient form by removing the time dependency of the
linear operator L0 (a), which is what makes it difficult to estimate the propagator.
The parametrization that follows is identical to that used in [13].
Rt
Recall that τ (t) := 0 λ−2 (s) ds for any τ ≥ 0, and a(τ ) := −λ(t(τ ))∂t λ(t(τ )).
Let t(τ ) be the inverse function of τ (t), and fix a (τ -timescale) constant T1 > 0. We
approximate λ(t(τ )) on the interval 0 ≤ τ ≤ T1 by a new gauge λ1 (t(τ )), chosen so
that λ1 (t(T1 )) = λ(t(T1 )), and α := −λ1 (t(τ ))∂t λ1 (t(τ )) = a(T1 ) is constant.
Then we introduce a new spatial variable z(x, t) := λ−1
1 (t)x and a new time
Rt
variable σ(t) := 0 λ−2
(s)
ds,
together
with
a
new
function
η(z, θ, σ) defined by
1
(B.4)
α
2
λ1 (t)e 4 z η(z, θ, σ) := λ(t)e
a(τ ) 2
4 y
ξ(y, θ, τ ) ≡ λ(t)φ(y, θ, τ ).
One should keep in mind that the variables z and y are related by z/y = λ(t)/λ1 (t).
Rt
R t −2
To relate the time scales σ(t) := 0 λ−2
(s) ds, we note that
1 (s) ds and τ = 0 λ
one may for all t(τ ) < T1 regard σ as a function of τ given by
Z t(τ )
(B.5)
σ(τ ) :=
λ−2
1 (s) ds.
0
Observing that the function σ is invertible, we denote its inverse by τ (σ).
Now we derive an equation for ∂θ η, which is related to ∂θ ξ by (B.4). One has
(B.6)
∂σ ∂θ η(σ) = −L0 (α) ∂θ η(σ) + D̃1 + D̃2 + D̃3 ,
where the operator L0 (α) is linear and autonomous,
L0 (α) := −∂z2 +
1
α2 2 α
z − − 1 − ∂θ2 .
4
2
2
The remaing terms on the rhs are
2
λ
2 − 2a
(B.7a)
D̃1 := − 2
+
1
−
α
∂θ η,
λ1 2 + b(τ (σ))y 2
λ1 − α z 2 a y 2
(B.7b)
D̃2 :=
e 4 e 4 D2 ,
λ
λ1 − α z 2 a y 2
(B.7c)
D̃3 :=
e 4 e 4 D3 .
λ
We estimate those terms in the following.
Lemma 15. If the assumptions of Section 6.1 hold, then there exists a constant C
such that one has
τ
≤ σ(τ ) ≤ Cτ,
C
λ1 (t)
(B.8)
|a − α| + − 1 ≤ Cβ(τ (t)),
λ(t)
40
ZHOU GANG AND DAN KNOPF
and
1
hyi
≤
≤C
C
hzi
(B.9)
in the time interval τ ∈ [0, T1 ]. Moreover,
(B.10)
3 X
−5 α4 z2 D̃k hzi e
L∞
k=1
13
5
.β
(τ (σ)).
This lemma is proved in Section B.2.
Now we apply Duhamel’s principle to the evolution equation (B.6), obtaining
Z σ(T1 )
3
X
∂θ η(σ(T1 )) = e−σ(T1 )L0 (α) ∂θ η0 +
e−(σ(T1 )−σ1 )L0 (α)
D̃k (σ1 ) dσ1 .
0
k=1
The definition of η implies that at time t = T1 , one has y = z, α = a(τ (T1 )) and
η = ξ. Together with the orthogonality conditions for ∂θ ξ in (B.2), this implies
that
∂θ η(σ(T1 )) = P7 ∂θ η(σ(T1 ))
( 3
)
Z σ(T1 )
X
−σL(α)
= P7 e
∂θ η0 +
e−(σ(T1 )−σ1 )L0 (α) P7
D̃k (σ1 ) dσ1
0
k=1
where P7 denotes orthogonal projection onto the subspace orthogonal to the seven
functions in (B.2).
To prove that η is sufficiently small, we use the following propagator estimates.
Lemma 16. Let the assumptions of Section 6.1 hold, Then for any time σ ≥ 0,
one has
αz 2
−5 αz4 2 −σL(α)
P7 [η]
. e−ασ hzi−5 e 4 η .
hzi e e
∞
∞
L
L
This lemma is proved in Section B.3 below.
Now we use estimate (B.10) to obtain
αz 2
−5 αz4 2
hzi e ∂θ η(σ(T1 )) ∞ . e−ασ hzi−5 e 4 ∂θ η(0) ∞
L
L
Z σ(T1 )
3
X
αz 2
+
e−ασ hzi−5 e 4
D̃k 0
−ασ
.e
β
k=1
21
10
σ(T1 )
Z
(0) +
e−ασ β
13
5
L∞
dσ
(τ (σ)) dσ
0
21
. β 10 (σ(T1 )).
21
Note that here we used the new condition khxi−5 ∂θ u0 kL∞ . b 10 (0) implied by
Assumption [A1] to get khyi−5 e
the proof by observing that
hzi−5 e
αz 2
4
a(0)y 2
4
21
∂θ ξ(0)kL∞ . β 10 (0). Finally, we complete
∂θ η(σ(T1 )) = hyi−5 e
ay 2
4
∂θ ξ(T1 ).
The desired estimate follows, and the proof of Lemma 11 is complete.
UNIVERSALITY IN MCF NECKPINCHES
41
B.1. Proof of Lemma 14.
Proof of Lemma 14. We start by proving the orthogonality condition. By (7.2), we
see that ξ is orthogonal to seven functions. By a simple integration by parts in θ,
we find that ∂θ ξ is also orthogonal to these same functions, which proves (B.2).
Next we turn to (B.3). Three observations make it relatively straightforward
to estimate the various terms there: (i) Most of the terms are of higher orders in
ξ, ∂y v, ∂θ v, and ∂y ∂θ v, which (appropriately weighted) are small in L∞ . (ii) The
presence of the operator ∂θ removes the slowly-changing θ-independent terms. And
(iii), the high weight hyi−5 here allows us to be generous when doing estimates.
We begin with the term D2 = ∂θ F = ∂θ [F1 (a, b) + F2 (a, b, β)]. The fact that
F1 is independent of θ makes ∂θ F = ∂θ F2 . Moreover the β0 -component, which
is also θ-independent, will vanish after applying ∂θ . Together with the estimates
16
16
∂τ βk − aβk = O(β 5 ), k = 1, 2, and ∂τ β` = O(β 5 ), ` = 3, 4, from (7.11) of
Lemma 9, this implies that
−5 ay42 . β3.
hyi e D2 L∞
For D1 , we relate e
β, obtaining
ay 2
4
ξ to v using (6.1), and we bound hyi−2
−5 ay42 hyi e D1 L∞
1
2 +a
2
−
1
2 +a
2+by 2
by
4
X
5
. β kv −2 ∂θ vkL∞ +
|βk | . β 2 .
k=1
Here, we used the estimate hyi−1 . v −1 from (A.11) and an assumption on v −2 |∂θ v|
from the outputs of the first bootstrap machine in Section 4.2.
We decompose D5 as
k
X
ay 2 −2 1 −3 3
−5
− v |∂θ v| +
|βk |
hyi ∂θ N3 e 4 . hyi−2 Va,b
2
k=1
k
X
−2
−3 −2
−2 3
+ hyi |v − Va,b | v |∂θ v| +
|βk |
k=1
h
1 + p2 i −2 2
+ hyi−3 ∂θ v −2
v |∂θ v|
1 + p2 + q 2
h
i
2pq
+ hyi−5 ∂θ v −1
∂θ ∂y v 2
2
1+p +q
h
i
q
+ hyi−5 ∂θ v −2
∂θ v 2
2
1+p +q
5
X
=
K` ,
l=1
with K` (` = 1, . . . , 5) naturally defined.
1
For K1 , we use the assumption A . β − 20 for the quantity A defined in (6.3) to
estimate the factor
1
−2 1 hyi−2 Va,b
− . hyi−2 | − a| + by 2 . β.
2
2
42
ZHOU GANG AND DAN KNOPF
33
By (6.9), the quantity v −2 |∂θ3 v| is bounded by β 20 . Combining these estimates with
the consequence |βk | . β 2 of Lemma 9 implies that
|K1 | . β
13
5
.
For K2 , we use ideas similar to those we used to control K1 to obtain
16
−5 ay42
hyi e K2 ∞ . β 5 .
L
−1
The presence of factors q = v ∂θ v and ∂θ ∂y v makes it relatively easy to control
the terms in K3 and K4 . Indeed, one may apply the estimates in (6.9) to obtain
−5 ay42
hyi e [K3 + K4 ] ∞ . β 3 .
L
The claimed estimate for D5 follows when one combines the estimates above. B.2. Proof of Lemma 15.
Proof of Lemma 15. Because the first three estimates are obtained exactly as in
the corresponding part of [13], we do not repeat those arguments here.
For (B.10), we begin by decomposing D̃1 into three terms,
h λ2 i
2a − 2α
b(τ (σ))y 2
2 − 2a
D̃1 = 1− 2
∂θ η+
∂θ η+ 2−2α
∂θ η.
2
2
λ1 2 + b(τ (σ))y
2 + b(τ (σ))y
2(2 + b(τ (σ))y 2 )
The factor by 2 in the last term on the rhs is controlled by hyi−2 by 2 ≤ b . β, which
together with the estimates for a − α and λ/λ1 − 1 in (B.8) implies that
α 2
a 2
−5 αz4 2 hyi e D̃1 ∞ . bhzi−3 e 4 z ∂θ η(σ) ∞ . β hyi−3 e 4 y ∂θ ξ(τ (σ)) ∞ .
L
L
L
Note that in the last step, we used (B.9) and the definition of η. Using (A.10),
which relates ∂θ ξ to ∂θ v, we get
51
−5 αz4 2 hyi e D̃1 ∞ . β 20 .
L
We relate the quantities D̃2 and D̃3 to D2 and D3 , respectively, using (B.8).
The we apply estimate (B.3) to obtain
a 2
−5 α4 z2
D̃k (σ) ∞ . hyi−5 e 4 y Dk (τ (σ)) ∞
hzi e
L
L
.β
13
5
The proof is complete.
(τ (σ)).
B.3. Proof of Lemma 16.
Proof of Lemma 16 . In our earlier work [13], we proved a similar result but with
the weight hzi−3 . To prove an estimate with the weight hzi−5 , we slightly generalize
a result proved in [12], which in turn is motivated by a result from [6].
In [12], the functions are θ-independent. So before adapting the arguments there,
we decompose η as
∞
X
eikθ fk (z),
η(z, θ) =
k=−∞
UNIVERSALITY IN MCF NECKPINCHES
e−ikθ η(z, θ) dθ. The explicit form of P7 allows us to write
X
P7 [η] = P3 [f0 ](z) + eiθ P2 [f1 ](z) + e−iθ P2 [f−1 ](z) +
eikθ fk (z)
1
2π
with fk (z) :=
(B.11)
43
R
S1
|k|>2
and
1
e−σL(α) P7 [η] = e−σL0 (α) P3 [f0 ](z) + eiθ e−σ[L0 (α)+ 2 ] P2 [f1 ](z)
X
1
k2
+ e−iθ e−σ[L0 (α)+ 2 ] P2 [f−1 ](z) +
eikθ e−σ[L0 (α)+ 2 ] fk (z).
|k|>2
αz 2
Here, P3 denotes orthogonal projection onto the subspace orthogonal to e− 4 ,
αz 2
αz 2
ze− 4 , and (αz 2 − 1)e− 4 , while P2 denotes orthogonal projection onto the subαz 2
αz 2
space orthogonal to e− 4 and ze− 4 .
In what follows, we focus on the first term on the rhs above; the remaining
terms are estimated similarly. To generalize Lemma 17 from [12], we note that in
the present situation, one has β = 0 and V = 0, which will make our analysis easier.
Following [12], we derive an integral kernel for e−σL0 (α) such that
Z ∞
2
α
(B.12)
e−σL0 (α) P3 [f0 ](z) = e 4 |z|
U0 (z, y)f (y) dy,
−∞
where
α
2
f (y) := e− 4 |y| P3 [f0 ](y),
and the integral kernel U0 is
1
U0 (z, y) := 4π(1 − e−2ασ )− 2
√
α e2ασ e
−
α(z−e−ασ y)2
2(1−e−2ασ )
.
To obtain the decay estimate, we integrate by parts in y three times, exploiting
the fact that ∂y U0 (z, y) has a factor e−ασ . To prepare for this, namely to ensure
that the functions we consider are in appropriate spaces, we use the fact that
2
2
2
α
α
α
P3 [f0 ] ⊥ {e− 4 |z| , ze− 4 |z| , z 2 e− 4 |z| } to see that f ⊥ {1, z, z 2 }, hence that
Z ∞
f (y) dy = 0,
−∞
Z ∞Z y
Z ∞
f (y1 ) dy1 dy =
yf (y) dy = 0,
(B.13)
−∞ −∞
−∞
Z ∞ Z y Z y1
Z ∞
f (y2 ) dy2 dy1 dy =
y 2 f (y) dy = 0.
−∞
−∞
−∞
−∞
Furthermore, there exists a constant C, independent of y, such that
|f (−3) (y)| ≤ Ce−
(B.14)
αy 2
2
(1 + |y|2 )khzi−5 e
αz 2
4
f0 kL∞ ,
where
f
(−3)
Z
y
Z
y1
Z
y2
Z
∞
Z
∞
Z
∞
f (y3 ) dy3 dy2 dy1 = −
(y) :=
−∞
−∞
−∞
f (y3 ) dy3 dy2 dy1 .
y
y1
y2
To obtain estimate (B.14), we repeatedly used the fact that the inequality
Z ∞
αy 2
αz 2
(1 + z)m e− 2 dz . (1 + y)m−1 e− 2
y
holds for y ≥ 0 and m ∈ R.
44
ZHOU GANG AND DAN KNOPF
The work that follows is different from [12]. Returning to (B.12), we integrate
by parts three times in y to find that
Z
αz 2
e−σL0 (α) P[f0 ](z) = −e 4
∂y3 U0 (z, y)f (−3) (y) dy.
Then we calculate directly and apply estimate (B.14) to obtain
(B.15)
−5 αz4 2 −σL0 (α)
P3 [f0 ](z)
hzi e e
Z
3
αz 2
e−3ασ
2 hzi−5
e
|z| + |y| + 1 U0 (z, y)f (−3) (y) dy
−2ασ
3
(1 − e
)
Z
2
3
αy 2
αz 2
e−3ασ
−5 αz
2 hzi−5 e 4 f .
|z| + |y| + 1 U0 (z, y)e− 2 1 + |y|2 dy
hzi
e
0
−2ασ
3
∞
(1 − e
)
L
Z
−3ασ
X
2
2
k
αy 2
e
−5 αz4
−k αz
2
.
e
hzi
e
f
U0 (z, y)e− 2 1 + |y| dy.
hzi
0
−2ασ
3
∞
(1 − e
)
L
.
k=1,2,3,4,5
Here in the second step, we use the simple observation
X
hzi−5 (|z| + |y| + 1)3 (1 + |y|2 ) .
hzi−k (1 + |y|)k
k=1,2,3,4,5
to show that
X
(B.16)
−k
hzi
e
αz 2
2
Z
U0 (z, y)e−
αy 2
2
1 + |y|
k
dy . e2ασ ,
k=1,2,3,4,5
Then we apply the same arguments as in the proof of Lemma 16 of [12], where the
cases k = 1, . . . , 4 were verified. Because this adaptation is straightforward and
simple, it does not need to be detailed here.
Now for σ ≥ 1, estimates (B.15) and (B.16) together imply the desired decay
estimate. On the other hand, for small σ, we apply (B.16) directly to (B.12) to
obtain a uniform bound. This completes the proof.
Appendix C. Proof of Lemma 12
Proof of Lemma 12. We first derive evolution equations for ξ± and (e−
To simplify notation, we introduce new functions ψ± defined by
ay2 ay 2
ψ± := e− 4 ∂y e 4 ξ± .
ay 2
4
∂y e
Next we make the following observations.
Lemma 17. The functions ξ± and ψ± satisfy the orthogonality conditions
n ay2
o
n ay2 o
ay 2
(C.1)
ξ± ⊥ e− 4 , ye− 4
and
ψ± ⊥ e− 4 ,
and evolve by
(C.2)
∂τ ξ± = − L0 (a)ξ± + e−
ay 2
4
[G1 + G2 ],
∂τ ψ± = − [L0 (a) + a]ψ± + e−
where
L0 (a) := −∂y2 +
ay 2
4
[G3 + G4 ],
a2 + ∂τ a 2 3a
y − .
4
2
ay 2
4
ξ)± .
UNIVERSALITY IN MCF NECKPINCHES
45
The terms Gk (k = 1, . . . , 4) on the rhs above satisfy the estimates
33
khyi−3 G2 kL∞ . β 20 ,
73
53
khyi−2 G4 kL∞ . β 20 .
khyi−2 G1 kL∞ . β 10 ,
63
khyi−1 G3 kL∞ . β 20 ,
The orthogonality conditions (C.1) follow directly from the corresponding properties for ξ. The remainder of the lemma is proved in Section C.1.
In what follows, we focus on estimating ξ+ . The remaining estimates are proved
similarly. For ξ+ , after going through the same procedures as we followed when
deriving (B.6), we obtain
(C.3)
∂σ [η+ ] = −L0 (α)[η+ ] + G̃1 (σ) + G̃2 (σ),
where L0 (α) is the autonomous linear operator
α2 2 3α
z −
,
4
2
and G̃k (k = 1, 2) are reparametrizations of Gk (k = 1, 2) defined similarly to (B.7).
The terms appearing in the evolution equation (C.3) satisfy the following estimates.
L0 (α) := −∂z2 +
Lemma 18. If the assumptions of Section 6.1 hold, then for any τ ≤ T1 and any
weight ` ≥ 0, one has
2
−` αz2
hzi e 4 η+ (σ) ∞ . hyi−` e ay4 ξ+ (τ (σ)) ∞ ,
(C.4)
L
L
along with
(C.5)
−2 33
hzi G̃1 ∞ . β 10
(τ (σ))
L
and
−3 73
hzi G̃2 ∞ . β 20
(τ (σ)).
L
The proofs of estimates (C.4) and (C.5) are almost identical to those that appear
in Lemma 9 above, hence are not repeated here.
Returning to equation (C.3), we apply Duhamel’s principle to obtain
Z σ
−σL0 (α)
(C.6)
η+ (σ) = e
η+ (0) +
e−(σ−σ1 )L0 (α) G̃1 (σ1 ) + G̃2 (σ1 ) dσ1 .
0
We again rely on a propagator estimate to prove the decay of η+ . Observe that the
quantum harmonic oscillator Lα has nonpositive eigenvalues with eigenfunctions
α 2
α 2
e− 4 z and ze− 4 z , which might make η grow. To control these eigenvectors, we
use the orthogonality properties of η+ .
Recall (B.5) and note that at time σ = σ(T1 ), namely τ = T1 , we have η+ (σ) =
2
2
ξ+ (T1 ) and eαz /4 = ea(τ )y /4 . Hence by (C.1), we have
P2 [η+ ](σ(T1 )) = η+ (σ(T1 )),
where P2 denotes the orthogonal projection onto the subspace orthogonal to the
2
2
span of {e−αz /4 , ze−αz /4 } (i.e. orthogonal to the unstable subspace of Lα ). We
apply P2 to both sides of (C.6), obtaining
η+ (σ(T1 )) = P2 e−σ(T )L0 (α) η+ (0)
Z σ(T1 )
(C.7)
+
e−(σ(T )−σ1 )L0 (α) P2 G̃1 (σ1 ) + G̃2 (σ1 ) dσ1
0
= A1 + A2 + A3 ,
where the terms A1 , A2 , A3 are naturally defined.
We estimate P2 e−σ(T ) L0 (α) as follows.
46
ZHOU GANG AND DAN KNOPF
Lemma 19. For all times that the assumptions of Section 6.1 hold, for any smooth
function g, any such time σ > 0, and any weight k = 2, 3, one has
αz 2 −k αz4 2 −σL0 (α) (C.8)
g ∞ . e−ασ hzi−k e 4 g ∞ ;
hzi e P2 e
L
L
11
− 10
while for the chosen weight hzi
, one has
−σL0 (α) αz 2
α
αz 2 11
− 11
(C.9)
g ∞ . e− 10 σ hzi− 10 e 4 g hzi 10 e 4 P2 e
L∞
L
.
Proof of Lemma 19. Both cases of Estimate (C.8) are proved by following the arguments in the proof of Lemma 16, mutatis mutandis.
Estimate (C.9) is obtained via an interpolation technique, as in Proposition 11.5
from [14]. Here, one interpolates between
αz 2
−1 αz4 2
hzi e P2 [e−σL0 (α) g] ∞ = hzi−1 e 4 P1 e−σL0 (α) P2 g ∞
L
L
−1 αz4 2 . hzi e g ∞
L
and the k = 2 case of estimate (C.8). Note that P1 denotes orthogonal projection
onto the subspace orthogonal to e−
αy 2
4
. We omit further details.
Continuing with the proof of Lemma 12, we now apply (C.4) to estimate the
first term A1 in the decomposition (C.7) by
11
αz 2
11
αz 2
α
− 10
e 4 A1 ∞ . e− 10 σ(T ) hzi− 10 e 4 η+ (0) ∞
hzi
L
L
ay 2
α
11
. e− 10 σ(T ) hyi− 10 e 4 ξ+ (0) ∞
L
.β
53
20
(T1 ).
11
53
Note that here we use the stronger13 restrictions khxi− 10 (u0 )± kL∞ . b 20 (0) =
53
11
53
53
β 20 (0) and khxi− 10 ∂x (u0 )± kL∞ . b 20 (0) = β 20 (0) on the initial data contained
11
in Assumption [A1] to get the estimate khyi− 10 e
11
estimate khyi− 10 ∂y e
(C.10)
ay 2
4
ay 2
4
53
ξ± (0)kL∞ . β 20 (0) and the
53
ξ± (0)kL∞ . β 20 (0). It follows that in the region βy 2 ≤ 20,
αz2 21
e 4 A1 . β 10
(T1 ).
Here we also used the fact that y = z at time τ = T1 .
To estimate the second term in the decomposition (C.7), we use (C.5), obtaining
Z σ(T )
1
33
33
−2 αz4 2 e− 8 (σ(T1 )−σ1 ) β 10 (τ (σ1 )) dσ . β 10 (T1 ).
hzi e A2 ∞ .
L
0
2
Hence in the region βy ≤ 20, one has
αz2 23
e 4 A2 . β 10
(C.11)
(T1 ).
To estimate the final term A3 in the decomposition (C.7), we use a different
norm: here we apply (C.5) again to get
Z σ(T1 )
1
19
19
−3 αz4 2 .
e− 8 (σ(T1 )−σ1 ) β 5 (τ (σ1 )) dσ . β 5 (T1 ).
hzi e A3 L∞
0
13That is to say, stronger than we needed in [13].
UNIVERSALITY IN MCF NECKPINCHES
47
Thus in the region βy 2 ≤ 20, one has
αz2 23
e 4 A3 . β 10
(T1 ).
(C.12)
Collecting estimates (C.10)–(C.12) above yields
a(τ )y2
αz2
23
4
ξ+ (T1 ) = e 4 η+ (σ(T1 )) . β 10 (T1 )
e
if βy 2 ≤ 20.
Because T1 ≥ 0 is arbitrary, this completes the proof of the estimate for ξ+ in
Lemma 12, modulo the proof of Lemma 17 that appears below. The remaining
estimates are obtained in a wholly analogous manner.
C.1. Proof of Lemma 17.
Proof of Lemma 17. The proof is in two parts.
Part (i) We start by deriving the first evolution equation in (C.2) and its
associated estimates. Instead of using equation (7.4) for ξ, it is more convenient to
work directly from equation (2.3) for v, in order to see and exploit certain fortuitous
cancellations.
Using notation defined in Section 2.1, we take an inner product heiθ , ·iS1 with
both sides of equation (2.3) to obtain
(C.13)
2π ∂τ v+ = ∂τ eiθ , v S1
= eiθ , F1 (p, q)∂y2 v S1 + eiθ , v −2 F2 (p, q)∂θ2 v S1
+ eiθ , v −1 F3 (p, q)∂θ ∂y v S1 + eiθ , v −2 F4 (p, q)∂θ v S1
− eiθ , ay∂y v S1 + eiθ , av S1 − eiθ , v −1 S1 .
To transform this equation into a form similar to (C.2), we have to allocate appropriate terms to G1 and G2 . For this purpose, we decompose various terms on the
rhs of equation (C.13).
We decompose the first term on the rhs of (C.13) into two terms,
iθ
e , F1 (p, q)∂y2 v S1 = eiθ , ∂y2 v S1 + eiθ , [F1 (p, q) − 1]∂y2 v S1
= 2π ∂y2 v+ + i eiθ , ∂θ [(F1 (p, q) − 1)∂y2 v] S1 ,
with the final term above obtained by integrating by parts in θ. Direct computation
yields
∂θ (F1 (p, q) − 1)∂y2 v = −
p2
∂θ ∂y2 v
1 + p2 + q 2
∂y ∂θ v∂y v 2
p2 (p∂θ p + q∂θ q) 2
−2
∂
v
+
2
∂ v.
y
1 + p2 + q 2
(1 + p2 + q 2 )2 y
48
ZHOU GANG AND DAN KNOPF
Now we group various terms on the rhs of (C.13) by introducing two functions
G̃1 and G̃2 , defined by
D
E
D
2
2∂y ∂θ v∂y v 2 E
iθ p (p∂θ p + q∂θ q) 2
2π G̃1 := eiθ , −
∂
v
∂
v
(C.14)
+
2
e
,
1 + p 2 + q 2 y S1
(1 + p2 + q 2 )2 y S1
iθ −2
+ e , v F2 (p, q)∂θ2 v S1 − eiθ , v −1 S1
+ eiθ , v −1 F3 (p, q)∂θ ∂y v S1 + eiθ , v −2 F4 (p, q)∂θ v S1
(C.15)
2π G̃2 := −
p2
∂θ ∂y2 v.
1 + p2 + q 2
This lets us write
(C.16)
∂τ v+ = ∂y2 − ay∂y + a v+ + G̃1 + G̃2 ,
where G̃1 and G̃2 satisfy the following estimates.
Lemma 20. For all times that the assumptions of Section 6.1 hold, one has
33
khyi−2 G̃1 kL∞ . β 10
and
73
khyi−3 G̃2 kL∞ . β 20 .
Proof of Lemma 20. We first estimate G̃2 , computing directly to find that
73
p2
−3
2 −2
(C.17)
∂
∂
v
|p|2 hyi−1 |∂θ ∂y2 v| . β 20 .
hyi
θ
y . hyi
2
2
1+p +q
In the final step above, we used the estimates hyi−1 . v −1 from (A.11) and
33
v −1 |∂θ ∂y2 v| . β 20 from (6.9), as well as the estimate
hyi−1 |p| ≡ hyi−1 |∂y v| ≤ hyi−1 |∂y Va,b |
(C.18)
+
4
X
11
|βk | + |∂y φ| ≤ β + β 10 M1,1 . β
k=0
implied by the assumptions on βk (k = 0, · · · , 4) and the assumption M1,1 . 1.
Now we turn to G̃1 . The assumptions in (6.9), which are outputs of the first
bootstrap machine, show that its first two terms admit the estimate
33
hyi−2 | · · · | . β 10 .
We treat the third and fourth terms of (C.13) together to exploit certain cancellations. By integrating by parts in θ, we obtain
iθ −2
e ,v F2 (p, q)∂θ2 v S1 − eiθ , v −1 S1
= −i eiθ , v −2 F2 (p, q)∂θ v S1 + i eiθ , v −2 ∂θ v S1 − eiθ , ∂θ [v −2 F2 (p, q)] ∂θ v S1
= −i eiθ , v −2 [F2 (p, q) − 1]∂θ v S1 − eiθ , ∂θ [v −2 F2 (p, q)] ∂θ v S1 .
2
1+p
Recall that F2 = 1+p
2 +q 2 . Controlling the terms that appear here is not hard, when
one takes advantage of the presence of the operator ∂θ , the presence of sufficiently
many factors of v −1 (helped by the estimate hyi−1 . v −1 ), and employs estimates
that have been proved and used frequently above. In this way, one finds that the
third and fourth terms are bounded by
33
khyi−2 (· · · )kL∞ . β 10 .
UNIVERSALITY IN MCF NECKPINCHES
49
In the same way, we find that the fifth and sixth terms of G̃1 also admit the estimate
33
khyi−2 (· · · )kL∞ . β 10 .
Collecting the estimates above, completes the proof.
Returning to the proof of Lemma 17, we go back to equation (C.16). We must
transform it further, because our objective is to derive an equation for ξ+ . The
decomposition of v in equation (6.1) implies that
ay 2
1
1
∂τ ξ+ = −L0 (a)ξ+ + e− 4 [a − ∂τ ]
β1 + β2
2
2i
ay 2
ay 2
ay 2
1
1
− ye− 4 ∂τ
β3 + β4 + e− 4 G̃1 + e− 4 G̃2 .
2
2i
ay 2
ay 2
To derive the simple form above, we repeatedly used the fact that e− 4 and ye− 4
are eigenvectors of L0 (a). By the equations for βk (k = 1, . . . , 4) in Lemma 9, we
have
33
[−∂τ + a][β1 − iβ2 ] + ∂τ [β3 − iβ4 ] . β 10
.
This leads us to write
∂τ ξ+ = −L0 (a)ξ+ + G1 + G2 ,
where
G1 := e−
ay 2
4
[−∂τ + a]
1
1
β 1 + β2
2
2i
− e−
ay 2
4
∂τ
ay 2
1
1
β3 y + β4 y + e− 4 G̃1 ,
2
2i
and
G2 := e−
ay 2
4
G̃2 .
To conclude the first part of the proof of Lemma 17, we apply Lemma 20 to obtain
33
73
−2 ay42 −3 ay42 and
hyi e G1 ∞ . β 10
hyi e G2 ∞ . β 20 .
L
L
Part (ii) We now derive the second evolution equation in (C.2) and prove its
associated estimates.
1
heiθ , ∂y viS1 .
Our first task is to derive an evolution equation for (∂y v)+ = 2π
(For (∂y v)− , it suffices to observe that ∂y v− = (∂y v)+ .) We compute directly to
get
∂τ ∂y v+ = ∂y3 v+ − ay∂y2 v+ + G̃3 + G̃4 ,
where
G̃3 := eiθ , [F1 (p, q) − 1]∂y3 v S1 + eiθ , ∂y F1 (p, q)∂y2 v S1
+ eiθ , v −2 [F2 (p, q) − 1]∂y ∂θ2 v S1 + eiθ , ∂y [v −2 F2 (p, q)]∂θ2 v S1
+ eiθ , ∂y [v −1 F3 (p, q)∂θ ∂y v] S1 + eiθ , ∂y [v −2 F4 (p, q)∂θ v] S1 .
and
G̃4 := heiθ , v −2 ∂θ2 ∂y viS1 + heiθ , v −2 ∂y viS1 .
We now proceed to estimate these quantities.
50
ZHOU GANG AND DAN KNOPF
To control G̃4 , we twice integrate by parts in θ, obtaining
G̃4 = ∂θ2 [v −2 eiθ ], ∂y v S1 + eiθ , v −2 ∂y v S1
= eiθ ∂θ2 v −2 + ieiθ ∂θ v −2 , ∂y v S1 .
Using the estimate khyi−1 ∂y vkL∞ = O(β) coming from (C.18) and the estimate
33
v −2 |∂θ2 v| = O(β 20 ) coming from assumption (6.9), which is an output of the first
bootstrap machine, we conclude that
53
khyi−1 G̃4 kL∞ . β 20 .
To control G̃3 , we combine assumptions in (6.8) and (6.9) with the estimate for
hyi−1 |∂y v| coming from (C.18) to get
63
khyi−2 G̃3 kL∞ . β 20 .
To conclude the second part of the proof, we again use the decomposition of v
in equation (6.1), this time to decompose the evolution equation for ∂y ψ+ . The
arguments used here are virtually identical to those which appear in part (i) of this
proof. Hence we omit further details.
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(Zhou Gang) California Institute of Technology
E-mail address: gzhou@caltech.edu
(Dan Knopf) University of Texas at Austin
E-mail address: danknopf@math.utexas.edu
URL: http://www.ma.utexas.edu/users/danknopf/